What is the sum from n=0 to infinity of n*2^{-n} ?

The sum from n=0 to infinity of n*2^{-n} is converges

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sum from n=0 to infinity of n*2^{-n}

n = 0 \sum \infty n \cdot 2^{- n}

converges

Solution steps

n = 0 \sum \infty n \cdot 2^{- n}

Apply Series Ratio Test:

converges

= converges

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What is the sum from n=0 to infinity of n*2^{-n} ?
The sum from n=0 to infinity of n*2^{-n} is converges