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

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

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

The series $$\sum_{n=1}^\infty a_n$$ 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
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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 $$\lim_{N \to \infty} s_N = b_1 - 0 = b_1$$.

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