Why Do We Need to Convert Series to Partial Fractions for Evaluation?

[trap101]

Now yesterday I got help in realizing how to evaluate the sums of certain series, but while doing it I never got the reason behind why we take a series such as: $\sum$ from k=1 to ∞ 1/k(k+3), I know how to solve the sum, but why do we have to convert it to a set of partial fractions in order to do it?

 

[DryRun]

It's the simplest way of finding the limit of its partial sums, by the telescoping series.

 

[trap101]

It's the simplest way of finding the limit of its partial sums, by the telescoping series.

ok, but what happens if the partial fraction that you end up with doesn't have a minus sign in it, then how would the telescoping occur?

 

[DryRun]

ok, but what happens if the partial fraction that you end up with doesn't have a minus sign in it, then how would the telescoping occur?

In that case, it would not be a telescoping series.

The definition of the telescoping series is that the limit of the partial sums must be equal to the sum of the first and last terms only.