Telescoping Series/Example 1

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

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

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

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

Proof
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:
 * $\displaystyle \lim_{N \mathop \to \infty} s_N = b_1 - 0 = b_1$

So:
 * $\displaystyle \sum_{n \mathop = 1}^\infty a_n = b_1$

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

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