Logan Wyatt
2022-07-04
Answered

Using Dirichlet series test I proved that the series $\sum _{n=2}^{\mathrm{\infty}}\frac{\mathrm{sin}nx}{n\mathrm{log}n}$ converges for all $x\in \mathbb{R}$

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asked 2022-01-21

A series is given as follows

$\frac{1}{6}+\frac{5}{6\cdot 12}+\frac{5\cdot 8}{6\cdot 12\cdot 18}+\frac{5\cdot 8\cdot 11}{6\cdot 12\cdot 18\cdot 24}+\dots$

asked 2022-03-28

Determining convergence of a series

$\sum _{n=0}^{\mathrm{\infty}}\sqrt{n}(\sqrt{{n}^{4}+1}-{n}^{2})$

asked 2022-07-13

Convergence of $\sum \frac{{a}_{n}}{\sqrt{n}}$ given that $\sum {a}_{n}^{2}$ converges

asked 2022-01-23

How to prove that

$\sum _{k\ge 1}\frac{1}{{2}^{k}{k}^{2}}=\frac{{\pi}^{2}}{12}-\frac{1}{2}{\mathrm{log}\left(2\right)}^{2}$

asked 2022-01-21

Prove that:

$x\mathrm{log}\left(x\right)=\left(\frac{x-1}{x}\right)+\frac{3}{2!}{\left(\frac{x-1}{x}\right)}^{2}+\frac{11}{3!}{\left(\frac{x-1}{x}\right)}^{3}+\dots \frac{{S}_{n}}{n!}{\left(\frac{x-1}{x}\right)}^{n}$

Where${S}_{n}=$ absolute Striling Numbers of first kind $(0,1,3,11,50,274\dots )$

Where

asked 2021-02-04

Calculate the following:

a. Find the Maclaurin series of

b. Find exactly the series of

asked 2022-02-28

What is the convergence rate of the tail of the series

$a}_{n}=\sum _{k>n}\text{exp}(-pk)k{\left(\mathrm{log}k\right)}^{2$