Write the statement of leibnitz test and prove it
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A series of the form
{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots \!}
where either all an are positive or all an are negative, is called an alternating series.
The alternating series test then says: if {\displaystyle |a_{n}|}decreases monotonically and {\displaystyle \lim _{n\to \infty }a_{n}=0}then the alternating series converges.
Moreover, let L denote the sum of the series, then the partial sum
{\displaystyle S_{k}=\sum _{n=1}^{k}(-1)^{n-1}a_{n}\!}
approximates L with error bounded by the next omitted term:
{\displaystyle \left|S_{k}-L\right\vert \leq \left|S_{k}-S_{k+1}\right\vert =a_{k+1}.\!}
{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}a_{n}=a_{0}-a_{1}+a_{2}-a_{3}+\cdots \!}
where either all an are positive or all an are negative, is called an alternating series.
The alternating series test then says: if {\displaystyle |a_{n}|}decreases monotonically and {\displaystyle \lim _{n\to \infty }a_{n}=0}then the alternating series converges.
Moreover, let L denote the sum of the series, then the partial sum
{\displaystyle S_{k}=\sum _{n=1}^{k}(-1)^{n-1}a_{n}\!}
approximates L with error bounded by the next omitted term:
{\displaystyle \left|S_{k}-L\right\vert \leq \left|S_{k}-S_{k+1}\right\vert =a_{k+1}.\!}
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