What is the sum from n=1 to infinity of ne^{-2n^2} ?

The sum from n=1 to infinity of ne^{-2n^2} is converges

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sum from n=1 to infinity of ne^{-2n^2}

n = 1 \sum \infty n e^{- 2 n^{2}}

converges

Solution steps

n = 1 \sum \infty n e^{- 2 n^{2}}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty arctan (\frac{1}{n})

k = 1 \sum \infty 3 (- \frac{1}{8})^{3 k}

n = 3 \sum \infty \frac{n - 1}{2 n + 1}

n = 2 \sum \infty (\frac{2}{5})^{2 n}

n = 1 \sum \infty \frac{2 n - 1}{2 n}

What is the sum from n=1 to infinity of ne^{-2n^2} ?
The sum from n=1 to infinity of ne^{-2n^2} is converges