and find the sum if it exists.

Sandra Allison
2022-01-18
Answered

Test the condition for convergence of

$\sum _{n=1}^{\mathrm{\infty}}\frac{1}{n(n+1)(n+2)}$

and find the sum if it exists.

and find the sum if it exists.

You can still ask an expert for help

David Clayton

Answered 2022-01-18
Author has **36** answers

Using Eulers

Karen Robbins

Answered 2022-01-19
Author has **49** answers

There is an alternate method and is as follows.

Notice that

where

This leads to the known result

alenahelenash

Answered 2022-01-24
Author has **343** answers

Alternatively, take
$\frac{1}{1-x}=\sum _{n=1}^{\mathrm{\infty}}{x}^{n-1}$
and integrate three times with lower limit 0, giving
$-\mathrm{log}(1-x)=\sum _{n=1}^{\mathrm{\infty}}\frac{{x}^{n}}{n}$
$x+(1-x)\mathrm{log}(1-x)=\sum _{n=1}^{\mathrm{\infty}}\frac{{x}^{n}}{n(n+1)}$
$\frac{3}{4}{x}^{2}-\frac{1}{2}x-\frac{1}{2}(1-x{)}^{2}\mathrm{log}(1-x)=\sum _{n=1}^{\mathrm{\infty}}\frac{{x}^{n}}{n(n+1)(n+2)}$

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I have to solve this recurrence using substitutions:

$(n+1)(n-2){a}_{n}=n({n}^{2}-n-1){a}_{n-1}-{(n-1)}^{3}{a}_{n-2}$ with ${a}_{2}={a}_{3}=1$

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For each natural number n and each number $x\in [0,1]$ let

$f}_{n}\left(x\right)=\frac{x}{nx+1$

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$\sum _{n=1}^{\mathrm{\infty}}\frac{n+\mathrm{log}\left(n\right)}{{(n+\mathrm{cos}\left(n\right))}^{3}}$

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