Telescoping Series/Example 1

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

Let $\left \langle {a_n} \right \rangle$ be a sequence whose terms are defined as:
 * $a_k = b_k - b_{k + 1}$

Then:
 * $\displaystyle \sum_{k \mathop = 1}^n a_k = b_1 - b_{n + 1}$

If $\left \langle {b_n} \right \rangle$ converges to zero, then:
 * $\displaystyle \sum_{k \mathop = 1}^\infty a_k = b_1$

Proof
If $\left \langle {b_k} \right \rangle$ 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:
 * $\displaystyle \sum_{k \mathop = 1}^\infty a_k = b_1$