What is the sum from n=1 to infinity of i/(5^n) ?

The sum from n=1 to infinity of i/(5^n) is 1/4 i

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sum from n=1 to infinity of i/(5^n)

n = 1 \sum \infty \frac{i}{5 ^{n}}

\frac{1}{4} i

Solution steps

i = 1 \sum \infty \frac{n}{5 ^{i}}

Apply the constant multiplication rule:

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

= i \cdot n = 1 \sum \infty \frac{1}{5 ^{n}}

n = k \sum \infty (a_{n}) = n = 0 \sum \infty (a_{n}) - a_{0} - a_{1} - \dots a_{k - 1}

= i (n = 0 \sum \infty \frac{1}{5 ^{n}} - \frac{1}{5 ^{0}})

n = 0 \sum \infty \frac{1}{5 ^{n}} = \frac{5}{4}

= i (\frac{5}{4} - \frac{1}{5 ^{0}})

Simplify

i (\frac{5}{4} - \frac{1}{5 ^{0}}) : \frac{1}{4} i

= \frac{1}{4} i

What is the sum from n=1 to infinity of i/(5^n) ?
The sum from n=1 to infinity of i/(5^n) is 1/4 i