What is the sum from k=1 to infinity of ke^{-7k^2} ?

The sum from k=1 to infinity of ke^{-7k^2} is converges

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sum from k=1 to infinity of ke^{-7k^2}

k = 1 \sum \infty k e^{- 7 k^{2}}

converges

Solution steps

k = 1 \sum \infty k e^{- 7 k^{2}}

Apply Series Ratio Test:

converges

= converges

n = 0 \sum \infty \frac{n !}{n}

n = 0 \sum \infty \frac{2 n}{n ^{3}}

n = 1 \sum \infty \frac{5 ^{n}}{n ^{2}}

n = 4 \sum \infty \frac{3}{n ^{2} - n}

k = 1 \sum \infty \frac{6 k ^{5}}{k !}

What is the sum from k=1 to infinity of ke^{-7k^2} ?
The sum from k=1 to infinity of ke^{-7k^2} is converges