Telescoping Series

Telescoping Series

Telescoping series get their name from the way terms tend to cancel out as you progress through the series. Here's a simple example of a telescoping series.

If we write out the terms, we have

Notice that this means we can easily sum up this series since so many terms cancel each other out. We are left with the fact that

Sometimes a telescoping series will not announce itself. You have to look for it.

Red Question: Find the following sum by re-writing the summand using partial fraction decomposition.

Green Question: (CEMC vol.8)

(i) Show that for any positive integer

(ii) Prove that

STEP 1998, Math II, #3: Show that the sum

of the first terms of the series

is

What is the limit of as ?

The numbers

are such that

and hence, or otherwise, evaluate when .

STEP 2003, Math III, #6: Show that

Hence, or otherwise, find all solutions of the equation

and are positive integers with .