Recent questions tagged sequence-series

Let $f_{n}(x)=\dfrac{1}{1+x^{n}}$. Pick the correct statement(s) from below.$f_{n}$ converges uniformly on $[0,1 / 2]$.$f_{n}$ converges uniformly on $[0,1)$.$f_{n}$ conv...

Identify the correct sequence of letters which will complete the following letter series.$\text{aabcd} - - - \text{bcdbc} - - - \text{dcdab} - - - $$\text{abaabccdd}$$\t...

A sequence of five natural numbers $s_{1} \leq s_{2} \leq s_{3} \leq s_{4} \leq s_{5}$ satisfy the following conditions: $\sum\limits_{i=1}^{5} s_{i}=35$$\sum\limits_{i=1...

For a positive integer $m$, let $d(m)$ denote the number of divisors of $m$, including $1$ and $m$. Define a sequence $\left\{a_{n}\right\}_{n=1}^{\infty}$ by\[a_{n}=\#\{...

Let $f: \mathbb{R} \longrightarrow \mathbb{R}$ be a continuous function such that $f(r)=f\left(r+\frac{1}{n}\right)$ for each $r \in \mathbb{Q}$ and each positive integer...

Let $a_1<a_2<a_3<a_4<\ldots<\ldots$ be a sequence of infinitely many positive integers which are in arithmetic progression. Furthermore, suppose $a_2, a_4, a_8$ are in ge...

A bug starts walking from the origin of a two dimensional plane. It walks 1 step up, $1 / 2$ step to the left then $1 / 4$ step down, then $1 / 8$ step to the right, then...

Let $f: \mathbb{R}_{\geq 0} \longrightarrow \mathbb{R}$ be the function\[f(x)=\left\{\begin{array}{ll}1, & x=0 \\x^{-x}, & x>0\end{array}\right.\]Determine whether the fo...

Let $f$ be a non-constant entire function with $f(0)=0$. Let $u$ and $v$ be the real and imaginary parts of $f$ respectively. Let $R>0$ and\[B=\sup \{u(z):|z|=R\}\]Show t...

 Let $a_{n}, n \geq 1$, be a sequence of positive real numbers such that $a_{n} \longrightarrow \infty$ as $n \longrightarrow \infty$. Then which of the following are tru...

Given a real number $\alpha \in(0,1)$, define a sequence $\left\{x_{n}\right\}_{n \geq 0}$ by the following recurrence relation:\[x_{n+1}=\alpha x_{n}+(1-\alpha) x_{n-1},...

Let $S_{n}=\sum_{k=1}^{n} \frac{1}{k}$. Then$S_{2^{n}} \geq \frac{n}{2}$ for every $n \geq 1$$S_{n}$ is a bounded sequence.$\left|S_{2^{n}}-S_{2^{n-1}}\right| \rightarrow...

If a sequence $\left\{f_{n}\right\}$ of continuous functions from $[0,1]$ to $\mathbb{R}$ converges uniformly on $(0,1)$ to a continuous function $f:[0,1] \rightarrow \ma...