3.11.E: Problems on Limits of Sequences (Exercises) - Mathematics

See also Chapter 2, §13.

Exercise (PageIndex{1})

Prove that if (x_{m} ightarrow 0) and if (left{a_{m} ight}) is bounded in (E^{1}) or (C,) then
a_{m} x_{m} ightarrow 0.
This is true also if the (x_{m}) are vectors and the (a_{m}) are scalars (or vice versa).
[Hint: If (left{a_{m} ight}) is bounded, there is a (K in E^{1}) such that
(forall m) quadleft|a_{m} ight|]
As (x_{m} ightarrow 0),
(forall varepsilon>0)(exists k)(forall m>k) quadleft|x_{m} ight|]
so (left|a_{m} x_{m} ight|

Exercise (PageIndex{2})

Prove Theorem 1(( ext { ii })).
[Hint: By Corollary 2(ii)(iii) in §14, we must show that (a_{m} x_{m}-a w ightarrow 0). Now
a_{m} x_{m}-a q=a_{m}left(x_{m}-q ight)+left(a_{m}-a ight) q.
where (x_{m}-q ightarrow 0) and (a_{m}-a ightarrow 0) by Corollary 2 of §14. Hence by Problem 1,
a_{m}left(x_{m}-q ight) ightarrow 0 ext { and }left(a_{m}-a ight) q ightarrow 0
(treat (q) as a constant sequence and use Corollary 5 in §14). Now apply Theorem 1((mathrm{i}) . ])

Exercise (PageIndex{3})

Prove that if (a_{m} ightarrow a) and (a eq 0) in (E^{1}) or (C,) then
(exists varepsilon>0)(exists k)(forall m>k) quadleft|a_{m} ight| geq varepsilon.
(We briefly say that the (a_{m}) are bounded away from (0,) for (m>k . )) Hence prove the boundedness of (left{frac{1}{a_{m}} ight}) for (m>k).
[Hint: For the first part, proceed as in the proof of Corollary 1 in (§14, ext { with } x_{m}=a_{m},) (p=a,) and (q=0 .)
For the second part, the inequalities
(forall m>k) quadleft|frac{1}{a_{m}} ight| leq frac{1}{varepsilon}
lead to the desired result. (])

Exercise (PageIndex{4})

Prove that if (a_{m} ightarrow a eq 0) in (E^{1}) or (C,) then
frac{1}{a_{m}} ightarrow frac{1}{a}.
Use this and Theorem 1(( ext { ii) to prove Theorem } 1( ext { iii), noting that })
frac{x_{m}}{a_{m}}=x_{m} cdot frac{1}{a_{m}}.
[Hint: Use Note 3 and Problem 3 to find that
(forall m>k) quadleft|frac{1}{a_{m}}-frac{1}{a} ight|=frac{1}{|a|}left|a_{m}-a ight| frac{1}{left|a_{m} ight|},
where (left{frac{1}{a_{m}} ight}) is bounded and (frac{1}{|a|}left|a_{m}-a ight| ightarrow 0 .) (Why?)
Hence, by Problem (1,left|frac{1}{a_{m}}-frac{1}{a} ight| ightarrow 0 .) Proceed. (])

Exercise (PageIndex{5})

Prove Corollaries 1 and 2 in two ways:
(i) Use Definition 2 of Chapter 2, §13 for Corollary (1(a),) treating infinite limits separately; then prove (b) by assuming the opposite and exhibiting a contradiction to ((a) .)
(ii) Prove (b) first by using Corollary 2 and Theorem 3 of Chapter 2, §13; then deduce (a) by contradiction.

Exercise (PageIndex{6})

Prove Corollary 3 in two ways (cf. Problem 5).

Exercise (PageIndex{7})

Prove Theorem 4 as suggested, and also without using Theorem 1((mathrm{i})).

Exercise (PageIndex{8})

Prove Theorem 2.
[Hint: If (overline{x}_{m} ightarrow overline{p},) then
(forall varepsilon>0)(exists q)(forall m>q) quad varepsilon>left|overline{x}_{m}-overline{p} ight| geqleft|x_{m k}-p_{k} ight| . quad(mathrm{Why} ?)
Thus by definition (x_{m k} ightarrow p_{k}, k=1,2, ldots, n).
Conversely, if so, use Theorem 1((mathrm{i})( ext { ii })) to obtain
sum_{k=1}^{n} x_{m k} vec{e}_{k} ightarrow sum_{k=1}^{n} p_{k} vec{e}_{k},
with (vec{e}_{k}) as in Theorem 2 of §§1-3].

Exercise (PageIndex{8'})

In Problem (8,) prove the converse part from definitions. (( ext { Fix } varepsilon>0, ext { etc. }))

Exercise (PageIndex{9})

Find the following limits in (E^{1},) in two ways: (i) using Theorem 1, justifying each step; (ii) using definitions only.
egin{array}{ll}{ ext { (a) } lim _{m ightarrow infty} frac{m+1}{m} ;} & { ext { (b) } lim _{m ightarrow infty} frac{3 m+2}{2 m-1}} { ext { (c) } lim _{n ightarrow infty} frac{1}{1+n^{2}} ;} & { ext { (d) } lim _{n ightarrow infty} frac{n(n-1)}{1-2 n^{2}}}end{array}
([ ext { Solution of }(mathrm{a}) ext { by the first method: Treat })
as the sum of (x_{m}=1) (constant) and
y_{m}=frac{1}{m} ightarrow 0 ext { (proved in } § 14 ).
Thus by Theorem 1((mathrm{i})),
frac{m+1}{m}=x_{m}+y_{m} ightarrow 1+0=1.
Second method: Fix (varepsilon>0) and find (k) such that
(forall m>k) quadleft|frac{m+1}{m}-1 ight|]
Solving for (m,) show that this holds if (m>frac{1}{varepsilon} .) Thus take an integer (k>frac{1}{varepsilon},) so
(forall m>k) quadleft|frac{m+1}{m}-1 ight|]
Caution: One cannot apply Theorem 1 (iii) directly, treating ((m+1) / m) as the quotient of (x_{m}=m+1) and (a_{m}=m,) because (x_{m}) and (a_{m}) diverge in (E^{1} .) (Theorem 1 does not apply to infinite limits.) As a remedy, we first divide the numerator and denominator by a suitable power of (m( ext { or } n) . ])

Exercise (PageIndex{10})

Prove that
left|x_{m} ight| ightarrow+infty ext { in } E^{*} ext { iff } frac{1}{x_{m}} ightarrow 0 quadleft(x_{m} eq 0 ight).

Exercise (PageIndex{11})

Prove that if
x_{m} ightarrow+infty ext { and } y_{m} ightarrow q eq-infty ext { in } E^{*},
x_{m}+y_{m} ightarrow+infty.
This is written symbolically as
" +infty+q=+infty ext { if } q eq-infty ."
Do also
" -infty+q=-infty ext { if } q eq+infty . "
Prove similarly that
"(+infty) cdot q=+infty ext { if } q>0"
"(+infty) cdot q=-infty ext { if } q<0."
[Hint: Treat the cases (q in E^{1}, q=+infty,) and (q=-infty) separately. Use definitions.]

Exercise (PageIndex{12})

Find the limit (or (underline{lim}) and (overline{lim})) of the following sequences in (E^{*} :)
(a) (x_{n}=2 cdot 4 cdots 2 n=2^{n} n !);
(b) (x_{n}=5 n-n^{3} ;)
(c) (x_{n}=2 n^{4}-n^{3}-3 n^{2}-1);
(d) (x_{n}=(-1)^{n} n !);
(e) (x_{n}=frac{(-1)^{n}}{n !}).
[Hint for ((mathrm{b}) : x_{n}=nleft(5-n^{2} ight) ;) use Problem 11.]

Exercise (PageIndex{13})

Use Corollary 4 in §14, to find the following:
(a) (lim _{n ightarrow infty} frac{(-1)^{n}}{1+n^{2}});
(b) (lim _{n ightarrow infty} frac{1-n+(-1)^{n}}{2 n+1}).

Exercise (PageIndex{14})

Find the following.
(a) (lim _{n ightarrow infty} frac{1+2+cdots+n}{n^{2}});
(b) (lim _{n ightarrow infty} sum_{k=1}^{n} frac{k^{2}}{n^{3}+1});
(c) (lim _{n ightarrow infty} sum_{k=1}^{n} frac{k^{3}}{n^{4}-1}).
[Hint: Compute (sum_{k=1}^{n} k^{m}) using Problem 10 of Chapter 2, §§5-6.]
What is wrong with the following "solution" of ((a) : frac{1}{n^{2}} ightarrow 0, frac{2}{n^{2}} ightarrow 0,) etc.; hence the limit is 0(?)

Exercise (PageIndex{15})

For each integer (m geq 0,) let
S_{m n}=1^{m}+2^{m}+cdots+n^{m}.
Prove by induction on (m) that
lim _{n ightarrow infty} frac{S_{m n}}{(n+1)^{m+1}}=frac{1}{m+1}.
[Hint: First prove that
(m+1) S_{m n}=(n+1)^{m+1}-1-sum_{i=0}^{m-1}left(egin{array}{c}{m+1} {i}end{array} ight) S_{m i}
by adding up the binomial expansions of ((k+1)^{m+1}, k=1, ldots, n . ])

Exercise (PageIndex{16})

Prove that
lim _{n ightarrow infty} q^{n}=+infty ext { if } q>1 ; quad lim _{n ightarrow infty} q^{n}=0 ext { if }|q|<1 ; quad lim _{n ightarrow infty} 1^{n}=1.
[Hint: If (q>1,) put (q=1+d, d>0 .) By the binomial expansion,
q^{n}=(1+d)^{n}=1+n d+cdots+d^{n}>n d ightarrow+infty . quad(mathrm{Why?})
If (|q|<1,) then (left|frac{1}{q} ight|>1 ;) so (lim left|frac{1}{q} ight|^{n}=+infty ;) use Problem (10 . ])

Exercise (PageIndex{17})

Prove that
lim _{n ightarrow infty} frac{n}{q^{n}}=0 ext { if }|q|>1, ext { and } lim _{n ightarrow infty} frac{n}{q^{n}}=+infty ext { if } 0]
[Hint: If (|q|>1,) use the binomial as in Problem 16 to obtain
|q|^{n}>frac{1}{2} n(n-1) d^{2}, n geq 2, ext { so } frac{n}{|q|^{n}}]
Use Corollary 3 with
x_{n}=0,left|z_{n} ight|=frac{n}{|q|^{n}}, ext { and } y_{n}=frac{2}{(n-1) d^{2}}
to get (left|z_{n} ight| ightarrow 0 ;) hence also (z_{n} ightarrow 0) by Corollary 2(( ext { iii) of } §14 . ext { In case } 0

Exercise (PageIndex{18})

Let (r, a in E^{1} .) Prove that
lim _{n ightarrow infty} n^{r} a^{-n}=0 ext { if }|a|>1.
[Hint: If (r>1) and (a>1,) use Problem 17 with (q=a^{1 / r}) to get (n a^{-n / r} ightarrow 0 .) As
obtain (n^{r} a^{-n} ightarrow 0).
If (r<1,) then (n^{r} a^{-n}

Exercise (PageIndex{19})

(Geometric series.) Prove that if (|q|<1,) then
lim _{n ightarrow infty}left(a+a q+cdots+a q^{n-1} ight)=frac{a}{1-q}.
aleft(1+q+cdots+q^{n-1} ight)=a frac{1-q^{n}}{1-q},
where (q^{n} ightarrow 0,) by Problem (16 . ])

Exercise (PageIndex{20})

Let (0[
lim _{n ightarrow infty} sqrt[n]{c}=1.
(left[ ext { Hint: If } c>1, ext { put } sqrt[n]{c}=1+d_{n}, d_{n}>0 . ext { Expand } c=left(1+d_{n} ight)^{n} ext { to show that } ight.)
so (d_{n} ightarrow 0) by Corollary (3 . ])

Exercise (PageIndex{21})

Investigate the following sequences for monotonicity, (underline{lim}), (overline{lim}), and (lim). (In each case, find suitable formula, or formulas, for the general term.)
(a) (2,5,10,17,26, ldots);
(b) (2,-2,2,-2, ldots);
(c) (2,-2,-6,-10,-14, ldots ;)
(d) (1,1,-1,-1,1,1,-1,-1, ldots ;)
(e) (frac{3 cdot 2}{1}, frac{4 cdot 6}{4}, frac{5 cdot 10}{9}, frac{6 cdot 14}{16}, ldots).

Exercise (PageIndex{22})

Do Problem 21 for the following sequences.
(a) (frac{1}{2 cdot 3}, frac{-8}{3 cdot 4}, frac{27}{4 cdot 5}, frac{-64}{5 cdot 6}, frac{125}{6 cdot 7}, ldots ;)
(b) (frac{2}{9},-frac{5}{9}, frac{8}{9},-frac{13}{9}, ldots ;)
(c) (frac{2}{3},-frac{2}{5}, frac{4}{7},-frac{4}{9}, frac{6}{11},-frac{6}{13}, ldots)
(d) (1,3,5,1,1,3,5,2,1,3,5,3, ldots, 1,3,5, n, ldots ;)
(e) (0.9,0.99,0.999, ldots);
(f) (+infty, 1,+infty, 2,+infty, 3, dots ;)
((mathrm{g})-infty, 1,-infty, frac{1}{2}, ldots,-infty, frac{1}{n}, ldots).

Exercise (PageIndex{23})

Do Problem 20 as follows: If (c geq 1,{sqrt[n]{c}} downarrow .(mathrm{Why} ?)) By Theorem (3,) (p=lim _{n ightarrow infty} sqrt[n]{c}) exists and
(forall n) quad 1 leq p leq sqrt[n]{c}, ext { i.e., } 1 leq p^{n} leq c .
By Problem (16, p) cannot be (>1,) so (p=1).
In case (0

Exercise (PageIndex{24})

Prove the existence of (lim x_{n}) and find it when (x_{n}) is defined inductively by
(i) (x_{1}=sqrt{2}, x_{n+1}=sqrt{2 x_{n}});
(ii) (x_{1}=c>0, x_{n+1}=sqrt{c^{2}+x_{n}});
(iii) (x_{1}=c>0, x_{n+1}=frac{c x_{n}}{n+1} ;) hence deduce that (lim _{n ightarrow infty} frac{c^{n}}{n !}=0).
[Hint: Show that the sequences are monotone and bounded in (E^{1}) (Theorem 3).
For example, in (ii) induction yields
Thus (lim x_{n}=lim x_{n+1}=p) exists. To find (p,) square the equation
x_{n+1}=sqrt{c^{2}+x_{n}} quad( ext { given })
and use Theorem 1 to get
p^{2}=c^{2}+p . quad(mathrm{Why?})
Solving for (p) (noting that (p>0 ),) obtain
p=lim x_{n}=frac{1}{2}left(1+sqrt{4 c^{2}+1} ight);
similarly in cases (i) and (iii). (])

Exercise (PageIndex{25})

Find (lim x_{n}) in (E^{1}) or (E^{*}) (if any), given that
(a) (x_{n}=(n+1)^{q}-n^{q}, 0(b) (x_{n}=sqrt{n}(sqrt{n+1}-sqrt{n}));
(c) (x_{n}=frac{1}{sqrt{n^{2}+k}});
(d) (x_{n}=n(n+1) c^{n},) with (|c|<1);
(e) (x_{n}=sqrt[n]{sum_{k=1}^{m} a_{k}^{n}},) with (a_{k}>0);
(f) (x_{n}=frac{3 cdot 5 cdot 7 cdots(2 n+1)}{2 cdot 5 cdot 8 cdots(3 n-1)}).
(a) (0(b) (x_{n}=frac{1}{1+sqrt{1+1 / n}},) where (1(c) Verify that
frac{n}{sqrt{n^{2}+n}} leq x_{n} leq frac{n}{sqrt{n^{2}+1}},
so (x_{n} ightarrow 1) by Corollary 3. (Give a proof.)
(d) See Problems 17 and 18.
(e) Let (a=max left(a_{1}, ldots, a_{m} ight) .) Prove that (a leq x_{n} leq a sqrt[n]{m} .) Use Problem (20 . ])
The following are some harder but useful problems of theoretical importance.
The explicit hints should make them not too hard.

Exercise (PageIndex{26})

Let (left{x_{n} ight} subseteq E^{1} .) Prove that if (x_{n} ightarrow p) in (E^{1},) then also
lim _{n ightarrow infty} frac{1}{n} sum_{i=1}^{n} x_{i}=p
(i.e., (p) is also the limit of the sequence of the arithmetic means of the (x_{n} ).)
[Solution: Fix (varepsilon>0 .) Then
(exists k)(forall n>k) quad p-frac{varepsilon}{4}]
Adding (n-k) inequalities, get
(n-k)left(p-frac{varepsilon}{4} ight)]
With (k) so fixed, we thus have
(forall n>k) quad frac{n-k}{n}left(p-frac{varepsilon}{4} ight)]
Here, with (k) and (varepsilon) fixed,
lim _{n ightarrow infty} frac{n-k}{n}left(p-frac{varepsilon}{4} ight)=p-frac{varepsilon}{4}.
Hence, as (p-frac{1}{2} varepsilon[
left(forall n>k^{prime} ight) quad p-frac{varepsilon}{2}]
left(exists k^{prime prime} ight)left(forall n>k^{prime prime} ight) quad frac{n-k}{n}left(p+frac{varepsilon}{4} ight)]
Combining this with (i), we have, for (K^{prime}=max left(k, k^{prime}, k^{prime prime} ight)),
left(forall n>K^{prime} ight) quad p-frac{varepsilon}{2}]
Now with (k) fixed,
lim _{n ightarrow infty} frac{1}{n}left(x_{1}+x_{2}+cdots+x_{k} ight)=0.
left(exists K^{prime prime} ight)left(forall n>K^{prime prime} ight) quad-frac{varepsilon}{2}]
Let (K=max left(K^{prime}, K^{prime prime} ight) .) Then combining with (ii), we have
(forall n>K) quad p-varepsilon]
and the result follows.

Exercise (PageIndex{26'})

Show that the result of Problem 26 holds also for infinite limits (p=pm infty in E^{*} .)

Exercise (PageIndex{27})

Prove that if (x_{n} ightarrow p) in (E^{*}left(x_{n}>0 ight),) then
lim _{n ightarrow infty} sqrt[n]{x_{1} x_{2} cdots x_{n}}=p.
[Hint: Let first (00,) use density to fix (delta>1) so close to 1 that
As (x_{n} ightarrow p),
(exists k)(forall n>k) quad frac{p}{sqrt[4]{delta}}]
Continue as in Problem (26,) replacing (varepsilon) by (delta,) and multiplication by addition (also subtraction by division, etc., as shown above). Find a similar solution for the case (p=+infty .) Note the result of Problem 20.]

Exercise (PageIndex{28})

Disprove by counterexamples the converse implications in Problems 26 and (27 .) For example, consider the sequences
1,-1,1,-1, dots
frac{1}{2}, 2, frac{1}{2}, 2, frac{1}{2}, 2, ldots

Exercise (PageIndex{29})

Prove the following.
(i) If (left{x_{n} ight} subset E^{1}) and (lim _{n ightarrow infty}left(x_{n+1}-x_{n} ight)=p) in (E^{*},) then (frac{x_{n}}{n} ightarrow p).
(ii) If (left{x_{n} ight} subset E^{1}left(x_{n}>0 ight)) and if (frac{x_{n+1}}{x_{n}} ightarrow p in E^{*},) then (sqrt[n]{x_{n}} ightarrow p).
Disprove the converse statements by counterexamples.
[Hint: For ((mathrm{i}),) let (y_{1}=x_{1}) and (y_{n}=x_{n}-x_{n-1}, n=2,3, ldots) Then (y_{n} ightarrow p) and
frac{1}{n} sum_{i=1}^{n} y_{i}=frac{x_{n}}{n},
so Problems 26 and (26^{prime}) apply.
For (ii), use Problem (27 .) See Problem 28 for examples. (])

Exercise (PageIndex{30})

From Problem 29 deduce that
(a) (lim _{n ightarrow infty} sqrt[n]{n !}=+infty);
(b) (lim _{n ightarrow infty} frac{n+1}{n !}=0);
(c) (lim _{n ightarrow infty} sqrt[n]{frac{n^{n}}{n !}}=e);
(d) (lim _{n ightarrow infty} frac{1}{n} sqrt[n]{n !}=frac{1}{e});
(e) (lim _{n ightarrow infty} sqrt[n]{n}=1).

Exercise (PageIndex{31})

Prove that
lim _{n ightarrow infty} x_{n}=frac{a+2 b}{3},
x_{0}=a, x_{1}=b, ext { and } x_{n+2}=frac{1}{2}left(x_{n}+x_{n+1} ight).
[Hint: Show that the differences (d n=x_{n}-x_{n-1}) form a geometric sequence, with ratio (q=-frac{1}{2},) and (x_{n}=a+sum_{k=1}^{n} d_{k} .) Then use the result of Problem (19 . ])

Exercise (PageIndex{32})

(Rightarrow 32 .) For any sequence (left{x_{n} ight} subseteq E^{1},) prove that
underline{lim} x_{n} leq underline{lim} frac{1}{n} sum_{i = 1}^{n} x_{i} leq overline{lim} frac{1}{n} sum_{i = 1}^{n} x_{i} leq overline{lim} x_{n} .
Hence find a new solution of Problems 26 and (26^{prime} .)
[Proof for (overline{lim}): Fix any (k in N .) Put
c=sum_{i=1}^{k} x_{i} ext { and } b=sup _{i geq k} x_{i}.
Verify that
(forall n>k) quad x_{k+1}+x_{k+2}+cdots+x_{n} leq(n-k) b.
Add (c) on both sides and divide by (n) to get
(forall n>k) quad frac{1}{n} sum_{i=1}^{n} x_{i} leq frac{c}{n}+frac{n-k}{n} b.
Now fix any (varepsilon>0,) and first let (|b|<+infty .) As (frac{c}{n} ightarrow 0) and (frac{n-k}{n} b ightarrow b,) there is (n_{k}>k) such that
left(forall n>n_{k} ight) quad frac{c}{n}]
Thus by (left(mathrm{i}^{*} ight)),
left(forall n>n_{k} ight) quad frac{1}{n} sum_{i=1}^{n} x_{i} leq varepsilon+b.
This clearly holds also if (b=sup _{i geq k} x_{i}=+infty .) Hence also
sup _{n geq n_{k}} frac{1}{n} sum_{i=1}^{n} x_{i} leq varepsilon+sup _{i geq k} x_{i}.
As (k) and (varepsilon) were arbitrary, we may let first (k ightarrow+infty,) then (varepsilon ightarrow 0,) to obtain
underline{lim} frac{1}{n} sum_{i=1}^{n} x_{i} leq lim _{k ightarrow infty} sup _{i geq k} x_{i}=overline{lim } x_{n} . quad( ext { Explain! }) ]

Exercise (PageIndex{33})

(Rightarrow 33 .) Given (left{x_{n} ight} subseteq E^{1}, x_{n}>0,) prove that
underline{lim} x_{n} leq underline{lim} sqrt[n]{x_{1} x_{2} cdots x_{n}} ext{ and } overline{lim} sqrt[n]{x_{1} x_{2} cdots x_{n}} leq overline{lim} x_{n} .
Hence obtain a new solution for Problem (27 .)
[Hint: Proceed as suggested in Problem (32,) replacing addition by multiplication.]

Exercise (PageIndex{34})

Given (x_{n}, y_{n} in E^{1}left(y_{n}>0 ight),) with
x_{n} ightarrow p in E^{*} ext { and } b_{n}=sum_{i=1}^{n} y_{i} ightarrow+infty,
prove that
lim _{n ightarrow infty} frac{sum_{i=1}^{n} x_{i} y_{i}}{sum_{i=1}^{n} y_{i}}=p.
Note that Problem 26 is a special case of Problem 34 (take all (y_{n}=1 )). [Hint for a finite (p :) Proceed as in Problem (26 .) However, before adding the (n-k) inequalities, multiply by (y_{i}) and obtain
left(p-frac{varepsilon}{4} ight) sum_{i=k+1}^{n} y_{i}]
(operatorname{Put} b_{n}=sum_{i=1}^{n} y_{i}) and show that
frac{1}{b_{n}} sum_{i=k+1}^{n} x_{i} y_{i}=1-frac{1}{b_{n}} sum_{i=1}^{k} x_{i} y_{i},
where (b_{n} ightarrow+infty( ext { by assumption }),) so
frac{1}{b_{n}} sum_{i=1}^{k} x_{i} y_{i} ightarrow 0 quad ext { (for a fixed } k ).
Proceed. Find a proof for (p=pm infty . ])

Exercise (PageIndex{35})

Do Problem 34 by considering (underline{lim}) and (overline{lim}) as in Problem 32.
(left[ ext { Hint: Replace } frac{c}{n} ext { by } frac{c}{b_{n}}, ext { where } b_{n}=sum_{i=1}^{n} y_{i} ightarrow+infty . ight])

Exercise (PageIndex{36})

Prove that if (u_{n}, v_{n} in E^{1},) with (left{v_{n} ight} uparrow) (strictly) and (v_{n} ightarrow+infty,) and if
lim _{n ightarrow infty} frac{u_{n}-u_{n-1}}{v_{n}-v_{n-1}}=p quadleft(p in E^{*} ight),
then also
lim _{n ightarrow infty} frac{u_{n}}{v_{n}}=p,
[Hint: The result of Problem (34,) with
x_{n}=frac{u_{n}-u_{n-1}}{v_{n}-v_{n-1}} ext { and } y_{n}=v_{n}-v_{n-1}.
leads to the final result. (])

Exercise (PageIndex{37})

From Problem 36 obtain a new solution for Problem (15 .) Also prove that
lim _{n ightarrow infty}left(frac{S_{m n}}{n^{m+1}}-frac{1}{m+1} ight)=frac{1}{2}.
[Hint: For the first part, put
u_{n}=S_{m n} ext { and } v_{n}=n^{m+1}.
For the second, put
u_{n}=(m+1) S_{m n}-n^{m+1} ext { and } v_{n}=n^{m}(m+1) . ]

Exercise (PageIndex{38})

Let (0[
a_{n+1}=sqrt{a_{n} b_{n}} ext { and } b_{n+1}=frac{1}{2}left(a_{n}+b_{n} ight), n=1,2, ldots
Then (a_{n+1}[
b_{n+1}-a_{n+1}=frac{1}{2}left(a_{n}+b_{n} ight)-sqrt{a_{n} b_{n}}=frac{1}{2}left(sqrt{b_{n}}-sqrt{a_{n}} ight)^{2}>0.
Deduce that
so (left{a_{n} ight} uparrow) and (left{b_{n} ight} downarrow .) By Theorem (3, a_{n} ightarrow p) and (b_{n} ightarrow q) for some (p, q in E^{1} .) Prove that (p=q,) i.e.,
lim a_{n}=lim b_{n}.
(This is Gauss's arithmetic-geometric mean of (a) and (b . ))
[Hint: Take limits of both sides in (b_{n+1}=frac{1}{2}left(a_{n}+b_{n} ight)) to get (q=frac{1}{2}(p+q) . ])

Exercise (PageIndex{39})

Let (0[
a_{n+1}=frac{2 a_{n} b_{n}}{a_{n}+b_{n}}, ext { and } b_{n+1}=frac{1}{2}left(a_{n}+b_{n} ight), quad n=1,2, ldots
Prove that
sqrt{a b}=lim _{n ightarrow infty} a_{n}=lim _{n ightarrow infty} b_{n}.
[Hint: Proceed as in Problem 38.]

Exercise (PageIndex{40})

Prove the continuity of dot multiplication, namely, if
overline{x}_{n} ightarrow overline{q} ext { and } overline{y}_{n} ightarrow overline{r} ext { in } E^{n}
(*or in another Euclidean space; see §9), then
overline{x}_{n} cdot overline{y}_{n} ightarrow overline{q} cdot overline{r}.

Fiitjee Maths DPP Modules pdf Download | Best FIITJEE maths dpps for IIT JEE | fiitjee maths dpp module pdf

Download FIITJEE Chapterwise DPP Level-I, Level-II and Level-III (Question Paper + Answer Key) for JEE Mains and Advanced Examination in PDF Free of Cost | Download fiitjee maths dpp module pdf

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Download Fiitjee RSM Honest Review by IITians | Details about FIITJEE RSM

FIITJEE DLP isn’t that smart for self study, the quality of queries square measure smart however the thought isn’t up to to the mark for JEE and conjointly the solutions provided aren’t up to the mark. If you’re going for iitjee then FIITJEE DLP are typically useful however if you are whole enthusiastic about self study then it’d be higher if you choose Resonance DLP package. If you’d wish to enter for check series of fiitjee then you will select FIITJEE Study material + check band however if you are getting separately then it’s not worthy.

Yes apart from Study Material, RTPF’s, GMP’s, JEE Archive, Olympiad Support Booklets square measure enclosed. Answers square measure progressing to be provided however they won’t be explained and second academics at fiitjee do entertain all doubts of students before and when categories however that is for sophistication area students solely.
During our time there accustomed be Exercise one then a pair of then among the tip there accustomed be miscellaneous queries then some theory question followed by Multiple exercise that accustomed covers all the classes of question asked in JEE like MCQ’s, Multiple selection multiple correct, whole number kind. queries were every which way unfold and other people weren’t sorted on the premise of problem. however they’ll have sorted them out into stage one and a combine of currently on the thought of problem.
Every Chapter has one module rather like the first module of physics of Scalar and Vector then second was Laws of Motion. each module can have one book. I don’t keep in mind the precise range of booklets however it had been 50+ Booklets for PCM combined.
Nah, schoolroom program is miles higher than RSM. In FIITJEE schoolroom study is everything they teach you, offer you with notes then the next day they raise you to unravel the precise queries in study material they tutored and thru the next category you will be able to raise the doubts then you get half time check, weekly check, Random category check. So, they created the study material in synchronize with their schoolroom program.
I would suggest Resonance Study Material and Fiitjee check series.

Separate modules for each chapter that contains illustrations, exercise between theories, resolved issues,assignment issues
Miscellaneous book of a particular phase(contains mixed question of 3–4 chapters)
chapter tests, section tests for jee mains and jee advanced
my suggestion is not to want dlp if you are in an exceedingly> very institute

  • ​Vectors pdf
  • ​AOD pdf
  • ​Area pdf
  • ​Binomial pdf
  • ​Circle pdf
  • ​Complex Numbers pdf
  • ​Determinants pdf
  • ​Differential Equations pdf
  • ​Ellipse pdf
  • ​Function2 pdf
  • ​Hyperbola pdf
  • ​Indefinite Integration pdf

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NCERT Exemplar Problems for Class 11 Maths

NCERT Exemplar Problems for Class 11 Maths PDF format to free download for new academic session 2021-2022 along with NCERT Solutions and offline apps 2021-22 based on latest CBSE Syllabus for new session.

These exemplar books are implemented into the syllabus to improve the mental ability and scientific temperament of the students. Completing your prescribed syllabus for the current year 2021-2022 students are required to practice the exemplar problems questions. The given examples are also important as per the exam point of view. It is better to do examplar problems books than so many other books like R D Sharma, Together with maths, R S Aggarwal, U – like, P K Garg, etc.

NCERT Exemplar Problems for Class 11 Maths

NCERT Exemplar Problems for Class 11 Maths in PDF

NCERT Exemplar Problems for Class 11 Maths are given below in PDF form to free download. Download NCERT Books and offline Apps based on latest CBSE Syllabus. Ask your doubts related to educational boards NIOS and CBSE through Discussion Forum. We are updating all the contents for current academic session 2021-2022. So, your feedback on updates are important for us. Just like last few years, please provide feedback and suggestions for 2021-22 also to improve the website.

NCERT Exemplar Problems for Class 11 Mathematics

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All the contents including NCERT Solutions, Test papers, Assignments, Chapter tests, etc., are being updated for new academic session 2021-22 on this website. We are going to increase the helping contents of NIOS as well as CBSE board. The Discussion Forum is being maintained to share the knowledge among with teachers and students. Everything is done just because of user’s feedback and suggestions. We are thankful to the suggestions received so far and expecting the same cooperation in future also.

Download NCERT Books and Offline Apps 2021-22 based on new CBSE Syllabus. Ask your doubts related to NIOS or CBSE Board and share your knowledge with your friends and other users through Discussion Forum.

Study time

Type Required
Lectures 30 sessions of 1 hour (20%)
Seminars 9 sessions of 1 hour (6%)
Private study 111 hours (74%)
Total 150 hours
Private study description

Weekly revision of lectures’ material
Solving non-assessed exercises (week 1, week 2, week 4, week 6, week 8)
Solving assessed problem sheets (week 3, week 5, week 7, week 9)
Preparation for exam (solving past exam papers)

Practice Problems

This page contains question sheets which are sent out to new students by many colleges before they arrive to start their undergraduate degree. These questions make suitable bridging material for students with single A-level Mathematics as they begin university - the material is partly revision, partly new material. All 11 sheets cover material relevant to the Mathematics, Mathematics & Statistics and Maths and Philosophy courses sheets 8, 9 and 10 are not relevant to the Mathematics and Computer Science degree.

For each sheet the subject matter is briefly described, and there is some recommended reading material the chapter numbers refer to the fourth edition of D.W.Jordan and P.Smith's book Mathematical Techniques, published by Oxford University Press in 2008.

  • Questions:
    • Sheet 1: Standard Functions and Techniques, PDF
      Reading: §§ 1.3, 1.6-1.8, 1.10-1.16
    • Sheet 2: Differentiation, PDF
      Reading: Chapter 2
    • Sheet 3: Further Differentiation, PDF
      Reading: §§ 3.1-3.5, 3.9-3.10
    • Sheet 4: Applications of Differentiation, PDF
      Reading: §§ 4.1-4.4
    • Sheet 5: Taylor Series, PDF
      Reading: §§ 5.1-5.4
    • Sheet 6: Complex Numbers, PDF
      Reading: Chapter 6
    • Sheet 7: Matrices, PDF
      Reading: Chapter 7
    • Sheet 8: Vectors, PDF
      Reading: §§ 9.1-9.4, 9.6
    • Sheet 9: The Scalar 'Dot' Product, PDF
      Reading: §§ 10.1-10.3, 10.9
    • Sheet 10: The Vector 'Cross' Product, PDF
      Reading: §§ 11.1-11.2
    • Sheet 11: Integration, PDF
      Reading: §§ 14.1-15.4, 15.8
    • All the above 11 sheets as one file: PDF
    • More challenging Questions:
      • Induction 1: PDF
        Reading: R.B.J.T. Allenby Numbers and Proof, Chapter 7
      • Induction 2: PDF
        Reading: R.B.J.T. Allenby Numbers and Proof, Chapter 7
      • Algebra 1: PDF
        Reading: No pre-requisites
      • Algebra 2: PDF
        Reading: Chapters 7 and 8
      • Calculus 1 - Curve Sketching: PDF
        Reading: §§ 4.1-4.4
      • Calculus 2 - Numerical Methods and Estimation: PDF
        Reading: §4.6, §5.2
      • Calculus 3 - Techniques of Integration: PDF
        Reading: §§17.5-17.7
      • Calculus 4 - Differential Equations: PDF
        Reading: §§ 22.3-22.4, Chapter 18
      • Calculus 5 - Further Differential Equations: PDF
        Reading: Chapter 19, §22.5
      • Complex Numbers: PDF
        Reading: Chapter 6
      • Geometry: PDF
        Reading: §10.1, §10.9, §11.1, §16.1
      • The second 11 sheets as one file: PDF
      • Further Sheets on Applied Mathematics (for Mathematics, Mathematics & Statistics students)
        • Dynamics 1 - Basic Definitions. Newton's Second Law PDF
        • Dynamics 2 - Oscillations and Further Examples. PDF

        Please contact us for feedback and comments about this page. Last update on 8 September 2019 - 15:03.

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        From English to Math

        Fatou's Lemma: Let $(X,Sigma,mu)$ be a measure space and $$ a sequence of nonnegative measurable functions. Then the function $displaystyle f_n>$ is measureable and $int_X liminf_ f_n dmu leq liminf_ int_X f_ndmu .$


        1st observation: $int g_k leq int f_n$ for all $ngeq k$. This follows easily from the fact that for a fixed $xin X$, $displaystyle>leq f_n(x)$ whenever $ngeq k$ (by definition of infimum). Hence $int displaystyle f_n> leq int f_n$ for all $ngeq k$, as claimed. This allows us to write egin int g_kleq inf_int f_n. qquad qquad (1) end

        2nd observation: $$ is an increasing sequence and $displaystyle g_k>=h$ pointwise. Thus, by the Monotone Convergence Theorem, egin intliminf_ f_n =int h = lim_ int g_k leq lim_ inf_int f_n = liminf_ int f_n end where the inequality in the middle follows from (1).

        Finally, $liminf f_n$ is measurable as we've proved before in the footnotes here.

        Exercise from Big Rudin

        The following is taken from chapter 1 of Rudin's Real and Complex Analysis. (Rudin, RCA, #1.8) Let $Esubset mathbb$ be Lebesgue measurable, and for $ngeq 0$ define $ f_n=egin chi_E & ext 1-chi_E & ext end $ What is the relevance of this example to Fatou's Lemma? For simplicity, let's just consider what happens when $X=[0,2]subset mathbb$ and we let $E=(1,2]subset X$. Then we get the following sequence of functions $f_n=egin chi_ <(1,2]>& ext chi_ <[0,1]>& ext. end $ The first few of these functions look like this:

        Notice that as $n$ increases, the graphs switch back and forth. For any given $n$, $int_<[0,2]>f_n=1$ but $liminf_nf_n=0$. (Recall that $liminf_n f_n$ is the infimum of all subsequential limits of $$). This shows us that $0=int_ <[0,2]>liminf_ f_n < liminf_int_ <[0,2]>f_n=1$ proving that a strict inequality in Fatou's Lemma is possible.

        3.11.E: Problems on Limits of Sequences (Exercises) - Mathematics

        For my younger students I usually start with a rule or function like 2x. We put a number in for x, get the number out, then put that output in for x, and continue that process. We get an infinite sequence of numbers. In this case the sequence diverges, doesn't go to a number. For example if we put 3->x, we get 6. We then put 6->x and we get 12.
        We get the infinite sequence 3, 6, 12, 24, .
        Later on, with older students, it is not a big step to use 1.1x as the function and show this is the same problem as increasing the population of a town 10% each year. A very important application.

        A teacher in one of Don's workshops, made up this function:. We'll pick a number, say 0, and put it in for x. What do we get out? 5 + 0/2 = 5. Then we put 5 in for x. What do we get out this time? 5 + 5/2 = 7.5 Now let's keep track of the infinite sequence we get: 0, 5, 7.5, 8.75, . The question is what's happening? Does this sequence converge? I ask my students to do the first 8 or so by hand, to make sure they can divide and write the answer as a fraction or mixed number and a decimal. Only then will I let them use a calulator to do more. Then I'll get them to the computer to use Mathematica to do 200 iterations and let it carry the answer to 100 decimal places!
        Finish the graph of this sequence, the beginning of which is shown below:

        Start with a new number, like 100 and see what happens.
        Start with -17 and see what happens. Graph these sequences on the same graph paper. Is there a pattern?
        Each infinite sequence has a limit of 10 for . Look at the numbers there. What do you think would happen if we started with 6 + x/2 ? a + x/2 ?
        What would happen with 5 + x/3 ? 5 + x/4 ?

        Another interesting function I do with my younger students is 6/x. Interesting things happen with this one!

        11 ways to solve a quadratic equation
        Method 1. By guessing and the sum and product of the roots (see above)
        Method 2. Solving x 2 - 5x + 6 = 0 for x to get x = .
        Jonathan, at age 7, solved this quadratic equation like this:

        2a.We can get an infinite continued fraction and find approximations of the roots of the equation
        2b.We'll iterate the function starting with different numbers, then graph these sequences.
        2c.Graph 3 successive 'pieces' of the infinite continued fraction
        2d.Graph y = , then connect points whose coordinates are consecutive input numbers
        Methods 3., 4., and 5. You solve x 2 - 5x + 6 = 0 for x, but in a different way than Jonathan did, (but not one of methods 6-11 below), and do the corresponding things as in 2a., 2b., 2c. and 2d above. You might find more than 3 other ways! Please let me know if you do.
        Method 6. Solving x 2 - x - 1 = 0 using a calculator to hone in on the two solutions.
        Method 7. By factoring (one of the 'normal' ways)
        Method 8. By completing the square
        Method 9. Using the quadratic formula
        Method 10. Graph x 2 - 5x + 6 = y (where it crosses the x-axis will be the roots, if they are real)
        Method 11. Spiraling in to the intersection of 2 curves
        Flash! This just happened (10/26/96): Colleen, a 7th grader, solved x 2 - x - 1 = 0 and got
        x = x 2 - 1. Try iterating this. It's exciting when something unexpected happens! That's what makes my teaching interesting and enjoyable. I've spent the last 2 hours working on this in Mathematica. To some answers to problems above from Ch. 8- part 2, iteration
        To problems from Ch. 8 part one- solving equations
        To order Don's materials
        To choose sample problems from other chapters
        Mathman Home

        Pure Maths

        All of the above topics will be coming to StudyWell in June 2021.

        • More binomial expansion, nth term.
        • Increasing, decreasing and periodic sequences.
        • Sigma notation.
        • Arithmetic sequences & series.
        • Geometric sequences & series.
        • Sequences in modelling.

        All of the above topics will be coming to StudyWell in July 2021.

        • arc length and area of a sector
        • small angle approximations
        • exact values of sin, cos and tan
        • reciprocal and inverse trigonometric functions
        • more trigonometric identities
        • double angle and compound angle formulae
        • trigonometric proof
        • problems in context

        All of the above topics will be coming to StudyWell in August 2021.

        • differentiate trigonometric functions from first principles, convex/concave functions
        • differentiate trigonometric and exponential functions
        • product rule, quotient rule and chain rule
        • implicit and parametric differentiation
        • construct simple differential equations

        All of the above topics will be coming to StudyWell from September 2021.

        • Integrate linear combinations, exponential and trigonometric functions.
        • Finding areas.
        • Understand that integration is the limit of a sum.
        • Integration by substitution and integration by parts.
        • Integrate using partial fractions.
        • Separation of variables.
        • Interpret the solution of a first order differential equation.

        All of the above topics will be coming to StudyWell from September 2021.

        • Approximate location of roots
        • Iterative methods
        • Newton-Raphson method
        • Numerical integration
        • Problems in context

        All of the above topics will be coming to StudyWell from September 2021.


        StudyWell is a website for students studying A-Level Maths (or equivalent. course). We have lots of resources including A-Level content delivered in manageable bite-size pieces, practice papers, past papers, questions by topic, worksheets, hints, tips, advice and much, much more.