What is the sum from k=2 to infinity of ke^{-3k^2} ?

The sum from k=2 to infinity of ke^{-3k^2} is converges

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sum from k=2 to infinity of ke^{-3k^2}

k = 2 \sum \infty k e^{- 3 k^{2}}

converges

Solution steps

k = 2 \sum \infty k e^{- 3 k^{2}}

Apply Series Ratio Test:

converges

= converges

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

n = 0 \sum \infty n^{2} e^{n}

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

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

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

What is the sum from k=2 to infinity of ke^{-3k^2} ?
The sum from k=2 to infinity of ke^{-3k^2} is converges