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Recent questions tagged convergence

0 votes
1 answer
1
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 $K \neq L$
asked Apr 24, 2018 in Calculus Tesla! 383 views
6 votes
2 answers
2
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
asked Dec 26, 2016 in Others jothee 350 views
3 votes
0 answers
3
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 $x \in \mathbb{R}$
asked Dec 20, 2015 in Set Theory & Algebra makhdoom ghaya 114 views
2 votes
0 answers
4
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 $\sum a_{n}^{2}$
asked Dec 20, 2015 in Set Theory & Algebra makhdoom ghaya 100 views
3 votes
0 answers
5
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.
asked Dec 19, 2015 in Set Theory & Algebra makhdoom ghaya 117 views
1 vote
0 answers
6
Let $f_{n}(x)$, for $n \geq 1$, be a sequence of continuous non negative functions on $[0, 1]$ such that $\displaystyle \lim_{n \rightarrow \infty} \int_{0}^{1} f_{n}(x) \text{d}x$ Which of the following statements is always correct? $f_{n} \rightarrow 0$ ... to $0$ 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
asked Dec 14, 2015 in Set Theory & Algebra makhdoom ghaya 109 views
1 vote
1 answer
7
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 $0$
asked Dec 14, 2015 in Set Theory & Algebra makhdoom ghaya 88 views
1 vote
0 answers
8
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.
asked Dec 9, 2015 in Calculus makhdoom ghaya 104 views
1 vote
0 answers
9
The series $\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}$ diverges.
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 98 views
1 vote
0 answers
10
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 $n \rightarrow \infty$
asked Dec 9, 2015 in Calculus makhdoom ghaya 122 views
1 vote
2 answers
11
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
asked Dec 9, 2015 in Set Theory & Algebra makhdoom ghaya 214 views
2 votes
1 answer
12
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. $\sum ^{\infty} _{i=0} b_{i}$ ... Statements $(2)$ and $(3)$ but not $(1)$. Statements $(1)$ and $(3)$ but not $(2)$. All the three statements. None of the three statements.
asked Nov 14, 2015 in Calculus makhdoom ghaya 251 views
2 votes
1 answer
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
asked Oct 15, 2015 in Calculus makhdoom ghaya 569 views
2 votes
0 answers
14
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
asked Oct 11, 2015 in Calculus makhdoom ghaya 214 views
2 votes
1 answer
15
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
asked Oct 11, 2015 in Calculus makhdoom ghaya 222 views
2 votes
2 answers
16
The series $\sum ^{\infty }_{n=1}\frac{(-1)^{n+1}}{\sqrt{n}}$ Converges but not absolutely. Converges absolutely. Diverges. None of the above.
asked Oct 11, 2015 in Quantitative Aptitude Arjun 197 views
1 vote
0 answers
17
One or more of the alternatives are correct. Marks will be given only if all the correct alternatives have been selected and no incorrect alternative is picked up. Which of the following improper integrals is (are) convergent? $\int ^{1} _{0} \frac{\sin x}{1-\cos x}dx$ ... $\int ^{1} _{0} \frac{1-\cos x}{\frac{x^5}{2}} dx$
asked Sep 13, 2014 in Calculus Kathleen 612 views
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