What is the sum from k=1 to infinity}e^{-sqrt(k of) ?

The sum from k=1 to infinity}e^{-sqrt(k of) is converges

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sum from k=1 to infinity}e^{-sqrt(k of)

k = 1 \sum \infty e^{- k}

converges

Solution steps

k = 1 \sum \infty e^{- k}

Simplify

e^{- k} : \frac{1}{e ^{k}}

= k = 1 \sum \infty \frac{1}{e ^{k}}

Apply Series Integral Test:

converges

= converges

x \to \infty lim (e^{x} - x)

3 v + \frac{d v}{d x} = 0

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What is the sum from k=1 to infinity}e^{-sqrt(k of) ?
The sum from k=1 to infinity}e^{-sqrt(k of) is converges