What is the sum from n=1 to infinity of n(1/3)^n ?

The sum from n=1 to infinity of n(1/3)^n is converges

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sum from n=1 to infinity of n(1/3)^n

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

converges

Solution steps

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

Apply Series Root Test:

converges

= converges

k = 1 \sum \infty \frac{k}{e ^{6 k}}

n = 1 \sum \infty \frac{n}{10 n - 3}

n = 0 \sum \infty 5 (\frac{5}{2})^{n}

n = 1 \sum \infty cos (\frac{4}{n})

n = 0 \sum \infty (\frac{1}{100})^{n}

What is the sum from n=1 to infinity of n(1/3)^n ?
The sum from n=1 to infinity of n(1/3)^n is converges