Articles of sequences and series

How to solve the following summation problem?

$$\sum\limits_{k=1}^n\arctan\frac{ 1 }{ k }=\frac{\pi}{ 2 }$$ Find value of $n$ for which equation is satisfied.

Prove or disprove: $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(-\infty,\infty)$

I’m trying to find out if $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(\infty,\infty)$. Here’s my attempt at an answer: If $ x\in(-\infty,0]:$ $$e^{-|x-n|}=e^{(-(x-n))}=e^{x-n}\le e^{-n}$$ $\sum e^{-n}$ converges, therefore $\sum_{n=0}^\infty e^{-|x-n|}$ converges uniformly in $(-\infty,0]$. If $x\in[0,\infty):$ Here no series $\sum a_n$ exists so that $e^{-|x-n|} \le a_n$ for every $x\in[0,\infty)$, because for every n, there exists […]

Period of a sequence defined by its preceding term

A sequence $x_n$ is defined such that $$x_{n+1}= \frac{\sqrt3 x_n -1}{x_n + \sqrt3}, n\ge1, x_0\neq-\sqrt3 $$ We now have to find the period of this sequence. By substituting values for $x_0$ I found out the period to be $6$. Also I proved this by finding all $x_{n+k}, 1\le k \le 6$ in terms of $x_n$ […]

I have to decide if $ \ell^1\subset c_0$ is closed or not.

I was asked to decide if $ \ell^1\subset c_0$ is closed or not, where $$\ell^1=\{(x_n)_{n\in\mathbb N}\subset\mathbb R:\sum_{n=0}^{\infty}|x_n|<\infty\}$$ $$c_0=\{ (x_n)_{n\in\mathbb N}\subset\mathbb R:\lim_{n\rightarrow\infty}x_n=0\}$$ In my opinion it is not closed so I want to prove my claim. Here is what I did, In order to show my claim I will try to show that the complement of […]

How do I get a sequence from a generating function?

For example if I have the generating function $\frac{1}{1-2x}$ then it corresponds to the sequence $1 + 2x + 4x^2 + 8x^3 +~…$. I know how to start from the sequence and get the generating function, but I don’t know how to start from the generating function and get the sequence. Similarly, what if I […]

Showing summation is bounded

I’m currently taking a Comp Sci class that is reviewing Calculus 2. I have a question: Show that the summation $\sum_{i=1}^{n}\frac{1}{i^2}$ is bounded above by a constant I realize that this question is already answered here Showing that the sum $\sum_{k=1}^n \frac1{k^2}$ is bounded by a constant Could anyone explain it to me further? I […]

Prove that if $\sum a_n$ converges, then $na_n \to 0$.

This question already has an answer here: If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$ [duplicate] 3 answers

Find the sum of the series $\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}-\cdots$

This question already has an answer here: Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} – \frac{1}{2} + \cdots $ 7 answers

Theorem 3.37 in Baby Rudin: $\lim\inf\frac{c_{n+1}}{c_n}\leq\lim\inf\sqrt{c_n}\leq\lim\sup\sqrt{c_n}\leq \lim\sup\frac{c_{n+1}}{c_n}$

Here’s Theorem 3.37 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: For any sequence $\{c_n\}$ of positive numbers, $$\lim_{n\to\infty} \inf \frac{c_{n+1}}{c_n} \leq \lim_{n\to\infty} \inf \sqrt[n]{c_n},$$ $$ \lim_{n\to\infty} \sup \sqrt[n]{c_n} \leq \lim_{n\to\infty} \sup \frac{c_{n+1}}{c_n}.$$ Now Rudin has given a proof of the second inequality. Here’s my proof of the first. Let $$\alpha […]

Prove $\sum_{k=1}^{\infty} \frac{\sin(kx)}{k} $ converges

How to prove $$\sum_{k=1}^{\infty} \frac{\sin(kx)}{k}$$ converges without using integral test?