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

The sum from n=0 to infinity of pie^{-n} is pi e/(e-1)

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

n = 0 \sum \infty π e^{- n}

π \frac{e}{e - 1}

Solution steps

n = 0 \sum \infty π e^{- n}

Apply the constant multiplication rule:

\sum c \cdot a_{n} = c \cdot \sum a_{n}

= π \cdot n = 0 \sum \infty e^{- n}

Simplify

e^{- n} : \frac{1}{e ^{n}}

= π \cdot n = 0 \sum \infty \frac{1}{e ^{n}}

Apply Series Geometric Test:

\frac{e}{e - 1}

= π \frac{e}{e - 1}

n = 0 \sum \infty (- 0.75)^{n}

n = 0 \sum \infty \frac{1}{5 - n}

n = 1 \sum \infty e^{n} + e^{- n}

n = 1 \sum \infty \frac{n}{7 n ^{2} + 1}

n = 0 \sum \infty \frac{5}{4 ^{n} - 3 ^{n}}

What is the sum from n=0 to infinity of pie^{-n} ?
The sum from n=0 to infinity of pie^{-n} is pi e/(e-1)