Telescoping Series/Example 2

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:

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

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

$=$

$\displaystyle \sum_{k \mathop = m}^n \left({b_k - b_{k - 1} }\right)$

$\displaystyle $

$=$

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

$\displaystyle $

$=$

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

$\displaystyle $

$=$

$\displaystyle \left({\sum_{k \mathop = m}^{n - 1} b_k + b_n}\right) - \left({b_{m - 1} + \sum_{k \mathop = m}^{n - 1} b_k}\right)$

$\displaystyle $

$=$

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

$\blacksquare$

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