What is the sum from n=1 to infinity of 5^ni ?

The sum from n=1 to infinity of 5^ni is diverges

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sum from n=1 to infinity of 5^ni

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

diverges

Solution steps

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

Apply the constant multiplication rule:

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

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

n = 0 \sum \infty 5^{n} =

diverges

If part of the expression diverges, then entire expression diverges

= diverges

n = 1 \sum \infty 5 (- \frac{1}{8})^{5 n}

n = 1 \sum \infty \frac{1}{n ^{2} + 25}

n = 4 \sum \infty (- 0.61)^{n}

n = 1 \sum \infty n (\frac{1}{3})^{n}

k = 1 \sum \infty \frac{k}{e ^{6 k}}

What is the sum from n=1 to infinity of 5^ni ?
The sum from n=1 to infinity of 5^ni is diverges