What is the sum from k=1 to infinity of k(1/2)^k ?

The sum from k=1 to infinity of k(1/2)^k is converges

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sum from k=1 to infinity of k(1/2)^k

k = 1 \sum \infty k (\frac{1}{2})^{k}

converges

Solution steps

k = 1 \sum \infty k (\frac{1}{2})^{k}

Apply Series Ratio Test:

converges

= converges

n = 0 \sum \infty \frac{4}{2 ^{2 n - 2}}

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

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}

What is the sum from k=1 to infinity of k(1/2)^k ?
The sum from k=1 to infinity of k(1/2)^k is converges