# Recent questions tagged convergence

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 $K \neq L$
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
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 $x \in \mathbb{R}$
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 $\sum a_{n}^{2}$
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.
1 vote
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
1 vote
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 $0$
1 vote
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.
1 vote
9
The series $\sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n}$ diverges.
1 vote
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 $n \rightarrow \infty$
1 vote
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
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.
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
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
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
The series $\sum ^{\infty }_{n=1}\frac{(-1)^{n+1}}{\sqrt{n}}$ Converges but not absolutely. Converges absolutely. Diverges. None of the above.
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? $\int ^{1} _{0} \frac{\sin x}{1-\cos x}dx$ ... $\int ^{1} _{0} \frac{1-\cos x}{\frac{x^5}{2}} dx$