Question

Justify if the series is convergent: sum_(n = 1)^oo e^(-n)

Series
ANSWERED
asked 2021-08-13
Justify if the series is convergent:
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{e}^{{-{n}}}\)

Expert Answers (1)

2021-08-14
\(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{e}^{{-{n}}}=\)
\(\displaystyle={\sum_{{{n}={1}}}^{\infty}}\frac{{1}}{{e}^{{n}}}\)
Here, common ratio, \(\displaystyle{a}=\frac{{1}}{{e}}\)
So, \(\displaystyle{S}=\frac{{\frac{{1}}{{e}}}}{{{1}–\frac{{1}}{{e}}}}=\frac{{\frac{{1}}{{e}}}}{{\frac{{{e}–{1}}}{{e}}}}\)
\(\displaystyle{S}=\frac{{1}}{{{e}–{1}}}\).
This series converges.
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