Telescoping Series/Examples

Examples of Telescoping Series

Example 1

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:

$\ds \sum_{k \mathop = 1}^n a_k = b_1 - b_{n + 1}$

Example 2

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:

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