Question

# Show that the series$\sum_{n=1}^{\infty} a_n$can be written in the telescoping form$\sum_{n=1}^{\infty}\left[\left(c-S_{n-1}\right)-\left(c-S_n\right)\right]$where $S_0=0$ and $S_n$ is the nth partial sum.

Solution

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Since the parcial sum is defined as $S_n=\displaystyle{\sum_{k=1}^{n}} a_k$ and $S_{n-1}=\displaystyle{\sum_{k=1}^{n-1}} a_k$ it means that

$S_n - S_{n-1}=\sum_{k=1}^{n} a_k - \sum_{k=1}^{n-1} a_k=a_n$

Hence,

$a_n =S_n - S_{n-1} = S_n -c+c- S_{n-1}=\left(c- S_n \right) - \left( c-S_{n-1}\right)$

Now let's rewrite the initial series

$\boldsymbol{\sum_{n=1}^{\infty} a_n } = \boldsymbol{\sum_{n=1}^{\infty} \left[\left(c - S_n \right) - \left(c - S_{n-1}\right) \right] }$

And this is the telescoping form of the series.

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