1. ## nth partial sum

Find a formula for the nth partial sum of the series and use it to find the seriesʹ sum if the series converges.

1)

I have no idea how to do this one. I missed it on the test and it might show up on the final

2. The 'key' is the identity...

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

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. i'm still not getting it

4. Originally Posted by krzyrice
Find a formula for the nth partial sum of the series and use it to find the seriesʹ sum if the series converges.

1)

I have no idea how to do this one. I missed it on the test and it might show up on the final
$\displaystyle \frac{7}{(n+1)(n+2)} = \frac{7}{n+1}-\frac{7}{n+2}$

Now,

$\displaystyle \sum_{n=1}^k \frac{7}{n+1}-\frac{7}{n+2} = \left( \frac{7}{2}-\frac{7}{3} \right) + \left( \frac{7}{3}-\frac{7}{4} \right)+....+ \left( \frac{7}{k+1}-\frac{7}{k+2}\right)$

$\displaystyle = \left( \frac{7}{2} -\frac{7}{k+2}\right)$

find the limit as k tends to infinity.

5. thanks makes sense now

6. Originally Posted by krzyrice
Find a formula for the nth partial sum of the series and use it to find the seriesʹ sum if the series converges.

1)

I have no idea how to do this one. I missed it on the test and it might show up on the final
Surely this is

$\displaystyle 7\sum_{n = 2}^{\infty}\frac{1}{n(n + 1)}$.

Using partial fractions:

$\displaystyle \frac{A}{n} + \frac{B}{n + 1} = \frac{1}{n(n + 1)}$

$\displaystyle \frac{A(n + 1) + Bn}{n(n + 1)} = \frac{1}{n(n + 1)}$

$\displaystyle A(n + 1) + Bn = 1$

$\displaystyle (A + B)n + A= 0n + 1$.

So $\displaystyle A = 1$ and $\displaystyle B = -1$.

Therefore we can write the sum as

$\displaystyle 7\sum_{n = 2}^{\infty}\frac{1}{n(n + 1)} = 7\sum_{n = 2}^{\infty}\left(\frac{1}{n} - \frac{1}{n + 1}\right)$

This is a telescoping series, so all terms except the first and last will cancel.

So the $\displaystyle n^{\textrm{th}}$ partial sum is

$\displaystyle 7\left(\frac{1}{2} - \frac{1}{n + 1}\right)$.

Therefore, the sum of the series is

$\displaystyle \lim_{n \to \infty}7\left(\frac{1}{2} - \frac{1}{n + 1}\right)$

$\displaystyle = 7\left(\frac{1}{2} - 0\right)$

$\displaystyle = \frac{7}{2}$.