Determine or find out if the following series converges or diverges.  If it converges find out its value.

Solution

We first require the partial sums for this series.

Here now, let's notice that we can make use of partial fractions on the series term to get,

1/  (i² + 3i +2)

= 1/ [(i+2) (i+1)]

=[1 / (i+1)] - [1 / (i+2)]

Previously you should be fairly adept at this as we spent a fair amount of time doing partial fractions back in the Integration Techniques.  If you have the requirement of a refresher you should go back and review that section.

 Thus, what does this do for us?  Well, let us start writing out the terms of the general partial sum for this series by using the partial fraction form.

Note that every term apart from the first term and last term cancelled out.  This is the main origin of the name telescoping series.

 This as well means that we can find out the convergence of this series by taking the limit of the partial sums.

Prove that the sequence of partial sums is convergent and thus the series is convergent and has a value of