is telescoping. Look at the partial sums:
because of cancellation of adjacent terms. So, the sum of the series, which is the limit of the partial sums, is 1.
You do have to be careful; not every telescoping series converges. Look at the following series:
You might at first think that all of the terms will cancel, and you will be left with just 1 as the sum.. But take a look at the partial sums:
This sequence does not converge, so the sum does not converge. This can be more easily seen if you simplify the expression for the term. You find that
and any infinite sum with a constant term diverges.
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