Calc6.60

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Discuss the convergence or divergence of the series \sum _{{k=4}}^{{\infty }}\left(e^{{-k+3}}-e^{{-k+1}}\right).

Let’s just look at the series by looking at the terms of this series.

\sum _{{k=4}}^{{\infty }}\left(e^{{-k+3}}-e^{{-k+1}}\right)=e^{{-1}}-e^{{-3}}+e^{{-2}}-e^{{-4}}+e^{{-3}}-e^{{-5}}+e^{{-4}}-e^{{-6}}+...\,

=e^{{-1}}+e^{{-2}}+e^{{-3}}-e^{{-3}}+e^{{-4}}-e^{{-4}}+e^{{-5}}-e^{{-5}}+...=e^{{-1}}+e^{{-2}}\,


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