How to evaluate what a series converges to?

[Chandasouk]

I was asked to evaluate the summation of \frac{1}{n(n+1)} from n=1 to infinity

I used partial fractions to obtain \frac{1}{n} - \frac{1}{n+1}

From here I don't understand how to evaluate. In my solutions manual, they plugged in values from 1 to infinity showing (1 - 1/2+ (1/2 - 1/3)...etc and created a new series called S_n = 1 - \frac{1}{n+1} then took the limit of that to infinity to get the answer 1.

How would I know what S_n should be?

 

[Mute]

So you want to evaluate:

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

One way to evaluate it is to consider what happens if you change variables in the second sum to m = n + 1. If you make this change of variables, do you see how you can deduce the solution?

 

[Dick]

Your series partial sum is (1-1/2)+(1/2-1/3)+...+(1/n-1/(n+1)). Regroup that like this 1+(-1/2+1/2)+(-1/3+1/3)+...+(-1/n+1/n)-1/(n+1). Lots of stuff cancels.

 

[Hurkyl]

How would I know what S_n should be?

Quick check -- you realize they are (presumably) using S_n for the n-th partial sum? And remember that an infinite sum is the limit of the partial sums?