Telescopic Summation for Finite Series

Telescopic summation is a more general technique for summing a series (finite or infinite) that does not fit the AP/GP/AGP mould. It expresses the sum of $n$ terms in just two terms — usually the first and the last — by making every intermediate term cancel. Once the middle cancels away, the far-off last term is brought right next to the first; hence the name "telescopic," like collapsing a telescope's sliding tubes.

The method. If the $k^{th}$ term $t_k$ can be written as a difference $t_k=f(k)-f(k+1)$, then

Example 5.17. For $\dfrac1{1+\sqrt2}+\dfrac1{\sqrt2+\sqrt3}+\dfrac1{\sqrt3+\sqrt4}+\cdots$, the $k^{th}$ term $t_k=\dfrac1{\sqrt k+\sqrt{k+1}}$ is rationalised: multiplying top and bottom by $\sqrt k-\sqrt{k+1}$ gives $t_k=\sqrt{k+1}-\sqrt k$, so the sum to $n$ terms telescopes to $\sqrt{n+1}-1$. …