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Recent questions tagged convergence
0
votes
1
answer
1
TIFR-2016-Maths-A: 2
Which of the following is false ? $\sum _{n=1}^{\infty } \sin \frac{1}{n}$ diverges $\sum _{n=1}^{\infty } \sin \frac{1}{n^{2}}$ converges $\sum _{n=1}^{\infty } \cos \frac{1}{n}$ diverges $\sum _{n=1}^{\infty } \cos \frac{1}{n^{2}}$ converges.
soujanyareddy13
asked
in
Calculus
Aug 30, 2020
by
soujanyareddy13
253
views
tifrmaths2016
convergence
0
votes
1
answer
2
ISI-2017-MMA-16
Let $(x_n)$ be a sequence of a real number such that the subsequence $(x_{2n})$ and $(x_{3n})$ converge to limit $K$ and $L$ respectively. Then $(x_n)$ always converge If $K=L$ then $(x_n)$ converge $(x_n)$ may not converge but $K=L$ it is possible to have $K \neq L$
Tesla!
asked
in
Calculus
Apr 24, 2018
by
Tesla!
765
views
isi2017
calculus
engineering-mathematics
non-gate
convergence
8
votes
2
answers
3
TIFR CSE 2016 | Part A | Question: 5
For a positive integer $N \geq 2$, let $A_N := \Sigma_{n=2}^N \frac{1}{n};$ $B_N := \int\limits_{x=1}^N \frac{1}{x} dx$ Which of the following statements is true? As $N \rightarrow \infty, \: A_N$ increases to infinity but $B_N$ ... $B_N < A_N < B_N +1$ As $N \rightarrow \infty, \: B_N$ increases to infinity but $A_N$ coverages to a finite number
go_editor
asked
in
Calculus
Dec 26, 2016
by
go_editor
1.1k
views
tifr2016
calculus
convergence
divergence
integration
non-gate
3
votes
1
answer
4
TIFR-2015-Maths-A-15
The series $\sum_{n=1}^{\infty}\frac{\cos (3^{n}x)}{2^{n}}$ Diverges, for all rational $x \in \mathbb{R}$ Diverges, for some irrational $x \in \mathbb{R}$ Converges, for some but not all $x \in \mathbb{R}$ Converges, for all $x \in \mathbb{R}$
makhdoom ghaya
asked
in
Set Theory & Algebra
Dec 20, 2015
by
makhdoom ghaya
336
views
tifrmaths2015
convergence
non-gate
2
votes
1
answer
5
TIFR-2015-Maths-A-11
Let $\left\{a_{n}\right\}$ be a sequence of real numbers. Which of the following is true? If $\sum a_{n}$ converges, then so does $\sum a_{n}^{4}$ If $\sum |a_{n}|$ converges, then so does $\sum a_{n}^{2}$ If $\sum a_{n}$ diverges, then so does $\sum a_{n}^{3}$ If $\sum |a_{n}|$ diverges, then so does $\sum a_{n}^{2}$
makhdoom ghaya
asked
in
Set Theory & Algebra
Dec 20, 2015
by
makhdoom ghaya
357
views
tifrmaths2015
convergence
non-gate
3
votes
0
answers
6
TIFR-2015-Maths-A-9
Let $\left\{a_{n}\right\}$ be a sequence of real numbers such that $|a_{n+1}-a_{n}|\leq \frac{n^{2}}{2^{n}}$ for all $n \in \mathbb{N}$. Then The sequence $\left\{a_{n}\right\}$ may be unbounded. The sequence $\left\{a_{n}\right\}$ is bounded but may not converge. The sequence $\left\{a_{n}\right\}$ has exactly two limit points. The sequence $\left\{a_{n}\right\}$ is convergent.
makhdoom ghaya
asked
in
Set Theory & Algebra
Dec 19, 2015
by
makhdoom ghaya
380
views
tifrmaths2015
convergence
non-gate
1
vote
0
answers
7
TIFR-2014-Maths-A-7
Let $f_{n}(x)$, for $n \geq 1$, be a sequence of continuous non negative functions on $[0, 1]$ such that $\displaystyle \lim_{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) \text{d}x$ ... $0$ point-wise $f_{n}$ will converge point-wise and the limit may be non-zero $f_{n}$ is not guaranteed to have a point-wise limit
makhdoom ghaya
asked
in
Set Theory & Algebra
Dec 14, 2015
by
makhdoom ghaya
320
views
tifrmaths2014
convergence
non-gate
1
vote
1
answer
8
TIFR-2014-Maths-A-5
Let $a_{n}=(n+1)^{100} e^{-\sqrt{n}}$ for $n \geq 1$. Then the sequence $(a_{n})_{n}$ is Unbounded Bounded but does not converge Bounded and converges to $1$ Bounded and converges to $0$
makhdoom ghaya
asked
in
Set Theory & Algebra
Dec 14, 2015
by
makhdoom ghaya
461
views
tifrmaths2014
convergence
non-gate
1
vote
0
answers
9
TIFR-2011-Maths-B-6
Suppose $f_{n}(x)$ is a sequence of continuous functions on the closed interval $[0, 1]$ converging to $0$ point wise. Then the integral $\int_{0}^{1} f_{n}(x) \text{d}x$ converges to 0.
makhdoom ghaya
asked
in
Calculus
Dec 9, 2015
by
makhdoom ghaya
285
views
tifrmaths2011
functions
convergence
1
vote
0
answers
10
TIFR-2011-Maths-A-24
The series $\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}$ diverges.
makhdoom ghaya
asked
in
Set Theory & Algebra
Dec 9, 2015
by
makhdoom ghaya
315
views
tifrmaths2011
convergence
1
vote
0
answers
11
TIFR-2011-Maths-A-6
The function $f_{n}(x)= n \sin (x/n)$ Does not converge for any $x$ as $n \rightarrow \infty$ Converges to the constant function $1$ as $n \rightarrow \infty$ Converges to the function $x$ as $n \rightarrow \infty$ Does not converge for all $x$ as $n \rightarrow \infty$
makhdoom ghaya
asked
in
Calculus
Dec 9, 2015
by
makhdoom ghaya
440
views
tifrmaths2011
convergence
1
vote
2
answers
12
TIFR-2011-Maths-A-1
Consider the sequence $\left \{x_{n} \right \}$ defined by $x_{n}=\frac{\left[nx\right]}{n}$ for $x \in \mathbb{R}$ where $[·]$ denotes the integer part. Then $\left \{x_{n} \right \}$ Converges to $x.$ Converges but not to $x.$ Does not converge Oscillates
makhdoom ghaya
asked
in
Set Theory & Algebra
Dec 9, 2015
by
makhdoom ghaya
602
views
tifrmaths2011
convergence
3
votes
1
answer
13
TIFR CSE 2014 | Part A | Question: 15
Consider the following statements: $b_{1}= \sqrt{2}$, series with each $b_{i}= \sqrt{b_{i-1}+ \sqrt{2}}, i \geq 2$, converges. $\sum ^{\infty} _{i=1} \frac{\cos (i)}{i^{2}}$ converges. $\sum ^{\infty} _{i=0} b_{i}$ ... $(3)$ but not $(1)$. Statements $(1)$ and $(3)$ but not $(2)$. All the three statements. None of the three statements.
makhdoom ghaya
asked
in
Calculus
Nov 14, 2015
by
makhdoom ghaya
638
views
tifr2014
convergence
non-gate
2
votes
1
answer
14
TIFR2010-Maths-B-13
Define $\left \{ x_{n} \right \}$ as $x_{1}=0.1,x_{2}=0.101,x_{3}=0.101001,\dots$ Then the sequence $\left \{ x_{n} \right \}$. Converges to a rational number Converges to a irrational number Does not coverage Oscillates
makhdoom ghaya
asked
in
Calculus
Oct 15, 2015
by
makhdoom ghaya
1.6k
views
tifrmaths2010
calculus
convergence
2
votes
0
answers
15
TIFR2010-Maths-B-5
If $f_{n}(x)$ are continuous functions from [0, 1] to [0, 1], and $f_{n}(x)\rightarrow f(x)$ as $n\rightarrow \infty $, then which of the following statements is true? $f_{n}(x)$ converges to $f(x)$ uniformly on [0, 1] $f_{n}(x)$ converges to $f(x)$ uniformly on (0, 1) $f(x)$ is continuous on [0, 1] None of the above
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
533
views
tifrmaths2010
calculus
convergence
2
votes
1
answer
16
TIFR2010-Maths-B-1
Let $U_{n}=\sin(\frac{\pi }{n})$ and consider the series $\sum u_{n}$. Which of the following statements is false? $\sum u_{n}$ is convergent $u_{n}\rightarrow 0$ as $n\rightarrow \infty $ $\sum u_{n}$ is divergent $\sum u_{n}$ is absolutely convergent
makhdoom ghaya
asked
in
Calculus
Oct 11, 2015
by
makhdoom ghaya
619
views
tifrmaths2010
calculus
convergence
2
votes
3
answers
17
TIFR2010-Maths-A-11
The series $\sum ^{\infty }_{n=1}\frac{(-1)^{n+1}}{\sqrt{n}}$ Converges but not absolutely. Converges absolutely. Diverges. None of the above.
Arjun
asked
in
Quantitative Aptitude
Oct 11, 2015
by
Arjun
810
views
tifrmaths2010
number-series
convergence
1
vote
0
answers
18
GATE CSE 1993 | Question: 02.2
The radius of convergence of the power series$\sum_{}^{\infty} \frac{(3m)!}{(m!)^3}x^{3m}$ is: _____________
Kathleen
asked
in
Calculus
Sep 13, 2014
by
Kathleen
1.2k
views
gate1993
calculus
convergence
normal
out-of-gate-syllabus
fill-in-the-blanks
3
votes
1
answer
19
GATE CSE 1993 | Question: 01.6
Which of the following improper integrals is (are) convergent? $\int ^{1} _{0} \frac{\sin x}{1-\cos x}dx$ $\int ^{\infty} _{0} \frac{\cos x}{1+x} dx$ $\int ^{\infty} _{0} \frac{x}{1+x^2} dx$ $\int ^{1} _{0} \frac{1-\cos x}{\frac{x^5}{2}} dx$
Kathleen
asked
in
Calculus
Sep 13, 2014
by
Kathleen
2.1k
views
gate1993
calculus
integration
convergence
out-of-gate-syllabus
multiple-selects
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