Telescoping series
In mathematics, a telescoping series is a series whose partial sums eventually only have a finite number of terms after cancellation. For example, let's try to find value of the series . We can see that
. Thus,
=
. Then, we can see that all of the terms except
and
. So the answer is
. We can see that in the process, we manipulated a large series so the many terms cancelled out with each other, leaving only a few terms that we could easily calculate with. This is usually how most telescoping series work.
Problems
Intermediate
- Find the value of
where
is the Riemann zeta function