A telescoping sum or series is a series where every term except the first few and the last few terms cancel out. This means that the series becomes very easy to turn into a closed expression.
It is not usually immediately obvious that a series will telescope, and usually some manipulation with partial fractions or other algebraic techniques may be necessary to make it a telescoping sum.
For example, consider the sum
Although it isn't a telescoping sum yet, we can use partial fractions on this sum. That is, break the terms into two fractions:
and by finding common denominators on the RHS, we see that `a=1` and `b=-1`.
This means that the sum now becomes
and rearranging the terms gives
and every term cancels out except the first two and the last two, so the final sum is just
so this means that