Calc6.59

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Discuss the convergence or divergence of the series \sum _{{k=0}}^{{\infty }}\left(e^{{-k}}-e^{{-(k+1)}}\right).

This is a telescopic series in a slightly different fashion than those we have looked at thus far.

\sum _{{k=0}}^{{\infty }}\left(e^{{-k}}-e^{{-(k+1)}}\right)=1-e^{{-1}}+e^{{-1}}-e^{{-2}}+e^{{-2}}-e^{{-3}}+...=1-\lim _{{k\rightarrow \infty }}e^{{-(k+1)}}=1


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