How would evaluate the following series?
I can get my calculator to evaluate this, but I am stuck when trying to calculate it manually. Is this a telescoping series where I would need to use partial fractions? I can't get it to work out that way.
These are nearly always able to be evaluated as a telescopic series after a partial fraction decomposition.
.
and .
Substituting into the second equation gives
.
Therefore .
So finally we have
.
Therefore our series is
since everything cancels except the first and last four terms
.