What is the sum from n=0 to infinity of n/(6^n) ?

The sum from n=0 to infinity of n/(6^n) is converges

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sum from n=0 to infinity of n/(6^n)

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

converges

Solution steps

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

Apply Series Ratio Test:

converges

= converges

n = 2 \sum \infty \frac{4}{n ( n - 1 )}

n = 1 \sum \infty \frac{1}{6 ^{n + 1}}

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

n = 1 \sum \infty \frac{n !}{6 ^{n}}

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

What is the sum from n=0 to infinity of n/(6^n) ?
The sum from n=0 to infinity of n/(6^n) is converges