Telescoping Series/Example 2

Theorem

Let $\sequence {b_n}$ be a sequence in $\R$.

Let $\sequence {a_n}$ be a sequence whose terms are defined as:

$a_k = b_k - b_{k - 1}$

Then:

$\ds \sum_{k \mathop = m}^n a_k = b_n - b_{m - 1}$

$\ds \sum_{k \mathop = m}^n a_k$

$=$

$\ds \sum_{k \mathop = m}^n \paren {b_k - b_{k - 1} }$

$\ds $

$=$

$\ds \sum_{k \mathop = m}^n b_k - \sum_{k \mathop = m}^n b_{k - 1}$

$\ds $

$=$

$\ds \sum_{k \mathop = m}^n b_k - \sum_{k \mathop = m - 1}^{n - 1} b_k$

$\ds $

$=$

$\ds \paren {\sum_{k \mathop = m}^{n - 1} b_k + b_n} - \paren {b_{m - 1} + \sum_{k \mathop = m}^{n - 1} b_k}$

$\ds $

$=$

$\ds b_n - b_{m - 1}$

$\blacksquare$

The term telescoping series arises from the obvious physical analogy with the folding up of a telescope.