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 = 2 \sum \infty \frac{n ^{n}}{n !}

k = 1 \sum \infty \frac{2}{k ln ( 9 k )}

n = 8 \sum \infty \frac{1}{n} - \frac{1}{n + 3}

n = 3 \sum \infty (1 + n)^{- 1}

n = 1 \sum \infty e^{- \frac{n}{3}}

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