What is the sum from n=0 to infinity of 8*(5/8)^n ?

The sum from n=0 to infinity of 8*(5/8)^n is 64/3

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sum from n=0 to infinity of 8*(5/8)^n

n = 0 \sum \infty 8 \cdot (\frac{5}{8})^{n}

\frac{64}{3}

Solution steps

n = 0 \sum \infty 8 (\frac{5}{8})^{n}

Apply the constant multiplication rule:

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

= 8 \cdot n = 0 \sum \infty (\frac{5}{8})^{n}

Apply Series Geometric Test:

\frac{8}{3}

= 8 \cdot \frac{8}{3}

Simplify

8 \cdot \frac{8}{3} : \frac{64}{3}

= \frac{64}{3}

n = 0 \sum \infty 5 + \frac{15}{5} + \frac{45}{16}

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

n = 2 \sum \infty (\frac{3}{7})^{n}

n = 0 \sum \infty (\frac{1}{n})^{2}

k = 0 \sum \infty \frac{2 ^{k}}{k !}

What is the sum from n=0 to infinity of 8*(5/8)^n ?
The sum from n=0 to infinity of 8*(5/8)^n is 64/3