What is the sum from n=1 to infinity of e^{-n/3} ?

The sum from n=1 to infinity of e^{-n/3} is converges

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sum from n=1 to infinity of e^{-n/3}

n = 1 \sum \infty e^{- \frac{n}{3}}

converges

Solution steps

n = 1 \sum \infty e^{- \frac{n}{3}}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty \frac{8 ^{n}}{n + 1}

n = 0 \sum \infty \frac{1}{( n + 2 ) n !}

n = 1 \sum \infty \frac{2 n + 1}{6 ^{n}}

n = 0 \sum \infty \frac{4}{3 - n}

n = 0 \sum \infty \frac{1}{n ^{4} + n ^{2}}

What is the sum from n=1 to infinity of e^{-n/3} ?
The sum from n=1 to infinity of e^{-n/3} is converges