What is the sum from n=0 to infinity of pi^{-3n} ?

The sum from n=0 to infinity of pi^{-3n} is converges

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sum from n=0 to infinity of pi^{-3n}

n = 0 \sum \infty π^{- 3 n}

converges

Solution steps

n = 0 \sum \infty π^{- 3 n}

Apply Series Ratio Test:

converges

= converges

n = 1 \sum \infty \frac{4 n}{n ^{2} + 1}

n = 0 \sum \infty 8 (\frac{2}{3})^{n}

n = 0 \sum \infty \frac{2}{n ^{3}}

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

n = 1 \sum \infty \frac{7}{n !}

What is the sum from n=0 to infinity of pi^{-3n} ?
The sum from n=0 to infinity of pi^{-3n} is converges