# Thread: Sum of series to infinite

1. ## Sum of series to infinite

$\sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)}$

2. ## Re: Sum of series to infinite

$\frac{u_{n+1}}{u_n}=\ldots=\frac{n}{n+3}=\frac{ \alpha n+\beta}{\alpha n+\gamma}$ (hipergeometric series) . According to a well known theorem , $S=\frac{u_1\gamma}{\gamma-(\alpha+\beta)}=\ldots=\frac{1}{4}$

3. ## Re: Sum of series to infinite

Originally Posted by FernandoRevilla
$\frac{u_{n+1}}{u_n}=\ldots=\frac{n}{n+3}=\frac{ \alpha n+\beta}{\alpha n+\gamma}$ (hipergeometric series) . According to a well known theorem , $S=\frac{u_1\gamma}{\gamma-(\alpha+\beta)}=\ldots=\frac{1}{4}$

4. ## Re: Sum of series to infinite

Originally Posted by BabyMilo
What methods have you covered, please?

5. ## Re: Sum of series to infinite

Originally Posted by FernandoRevilla
What methods have you covered, please?

I havent been taught how to do this really.
I have covered the test for convergent but the book asks for this.
I used partial fraction which they all diverges.

6. ## Re: Sum of series to infinite

Note that

$\frac{1}{n(n+1)(n+2)} = \frac{1}{2}\left(\frac{1}{n} - \frac{1}{n+1}\right) - \frac{1}{2}\left(\frac{1}{n+1} - \frac{1}{n+2}\right)$

Write some terms in the series

$\sum_{n=1}^{\infty} \frac{1}{2}\left(\frac{1}{n} - \frac{1}{n+1}\right) - \frac{1}{2}\left(\frac{1}{n+1} - \frac{1}{n+2}\right)$

and see what happens.