What is the sum from n=0 to infinity of 3^n*7^{-n} ?

The sum from n=0 to infinity of 3^n*7^{-n} is 7/4

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sum from n=0 to infinity of 3^n*7^{-n}

n = 0 \sum \infty 3^{n} \cdot 7^{- n}

\frac{7}{4}

Solution steps

n = 0 \sum \infty 3^{n} \cdot 7^{- n}

Simplify

7^{- n} : \frac{1}{7 ^{n}}

= n = 0 \sum \infty 3^{n} \cdot \frac{1}{7 ^{n}}

Simplify

3^{n} \cdot \frac{1}{7 ^{n}} : (\frac{3}{7})^{n}

= n = 0 \sum \infty (\frac{3}{7})^{n}

Apply Series Geometric Test:

\frac{7}{4}

= \frac{7}{4}

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

n = 1 \sum \infty \frac{2}{3 n + 7}

n = 0 \sum \infty (\frac{10 π}{e})^{n}

k = 1 \sum \infty \frac{k !}{10 5 ^{k}}

What is the sum from n=0 to infinity of 3^n*7^{-n} ?
The sum from n=0 to infinity of 3^n*7^{-n} is 7/4