Telescoping Series

Theorem

Let $\left \langle {a_n} \right \rangle$ be a sequence in $\R$.

Suppose that each $a_k$ can be expressed as the difference between two terms $a_k = b_k - c_k$ such that $c_k = b_{k+1}$.

Let $\left \langle {s_N} \right \rangle$ be the sequence of partial sums of the series $\displaystyle \sum_{n=1}^\infty a_n$.

Then $s_N = b_1 - b_{N+1}$.

If $\left \langle {b_n} \right \rangle$ converges to zero, then $\displaystyle \sum_{n=1}^\infty a_n = b_1$.

The series $\displaystyle \sum_{n=1}^\infty a_n$ is known as a telescoping series from the obvious physical analogy of the folding up of a telescope.

The technique of preparing the terms into this format is also known as the method of differences.

Proof

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle s_N\)	\(=\)	\(\displaystyle \sum_{n=1}^N a_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n=1}^N \left({b_n - c_n}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n=1}^N \left({b_n - b_{n+1} }\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n=1}^N b_n - \sum_{n=1}^N b_{n+1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \sum_{n=1}^N b_n - \sum_{n=2}^{N+1} b_n\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	by Permutation of Indices
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle b_1 + \sum_{n=2}^N b_n - \sum_{n=2}^N b_n - b_{N+1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle b_1 - b_{N+1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

Thus $s_N = b_1 - b_{N+1}$.

If $\left \langle {b_n} \right \rangle$ converges to zero, then $b_{N+1} \to 0$ as $N \to \infty$.

Thus:

$\displaystyle \lim_{N \to \infty} s_N = b_1 - 0 = b_1$

So:

$\displaystyle \sum_{n=1}^\infty a_n = b_1$

$\blacksquare$

Notes

Sometimes the word concertina is used in this context, but this is an even more informal usage.