Telescoping Series/Examples

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Examples of Telescoping Series

Example 1

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:

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


Example 2

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:

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