What is the sum from k=1 to infinity of k^3e^{-2k} ?

The sum from k=1 to infinity of k^3e^{-2k} is converges

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sum from k=1 to infinity of k^3e^{-2k}

k = 1 \sum \infty k^{3} e^{- 2 k}

converges

Solution steps

k = 1 \sum \infty k^{3} e^{- 2 k}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty (- 1.99 + 1)^{n}

n = 1 \sum \infty \frac{2}{n ^{10}}

n = 0 \sum \infty \frac{n ^{6}}{n !}

n = 4 \sum \infty 6^{- n}

n = 0 \sum \infty n \cdot 2

What is the sum from k=1 to infinity of k^3e^{-2k} ?
The sum from k=1 to infinity of k^3e^{-2k} is converges