1. ## Determine the convergence.

For the following series ∑ of a(n) between n=1 and ∞, determine if
the series converges and if so find the limit. (Hint: use partial
fractions to express a(n).)

(a) a(n)=1/(n(n+1))

(b) a(n)=1/(n^2+2n)

(c) a(n)=1/(n(n^2-1))
(As a(1) is not defined consider ∑ of a(n) between n=2 and ∞.)

2. #1:

Are you familiar with telescoping sums?.

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

$=\sum_{1}^{\infty}\left[\frac{1}{n}-\frac{1}{n+1}\right]$

$=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{4}-\frac{1}{5}\right)+......$

Notice what is going on?.

Everything cancels one another and you're left with the 1 in the very beginning. So, it converges to 1.

The others may be similar. Check and see.

3. Originally Posted by galactus
$\sum_{1}^{\infty}\frac{1}{n(n+1)}$
Just for fun, I just solve this one as follows:

$\sum\limits_{n = 1}^\infty {\frac{1}
{{n(n + 1)}}} = \int_0^1 {\left\{ {\sum\limits_{n = 1}^\infty {\frac{{x^n }}
{n}} } \right\}\,dx} = - \int_0^1 {\ln (1 - x)\,dx} = \int_0^1 \!{\int_0^{\,x} {\frac{{dy\,dx}}
{{1 - y}}} } .$

So

$\sum\limits_{n = 1}^\infty {\frac{1}
{{n(n + 1)}}} = \int_0^1 \!{\underbrace {\int_y^1 {dx} }_{1\, -\, y}\,\frac{{dy}}
{{1 - y}}} = 1,$

as required.

4. Ok i did part b and got it converges to 3/4

because $1/(n^2+2n)= (1/2n)-(1/2n+4)$

5. c.
You can compare with $\zeta(s)=\sum_{n=1}^{\infty}{\frac{1}{n^s}}$

This series converges if $s>1$

We have (n>1): $n^2-1 ==> $n\cdot{(n^2-1)} so $\frac{1}{n^3}<\frac{1}{n\cdot{(n^2-1)}}$

Thus it follows that the series converges by comparison with $\zeta(3)$

To calculate the sum:
$\frac{1}{n\cdot{(n^2-1)}}=\left(\frac{1}{n+1}\right)\cdot{\left(\frac{1 }{n-1}-\frac{1}{n}\right)}$

So; $\frac{1}{n\cdot{(n^2-1)}}=\frac{1}{(n+1)\cdot(n-1)}-\frac{1}{(n+1)\cdot(n)}$

Summing: $\sum_{n=2}^{\infty}\frac{1}{n\cdot{(n^2-1)}}=\sum_{n=2}^{\infty}\frac{1}{(n+1)\cdot(n-1)}-\sum_{n=2}^{\infty}\frac{1}{(n+1)\cdot{n}}$

Both series on the right are telescoping series

6. Ok i got

a) 1

b) 3/4

c) 1/4

can any1 tell me if there right.

And is there in index of how to use the math tags for this forum [tex] those.