Write out the first few terms of the series∑n=0∞(−1)n5nWhat is the series' sum?

Expert answers to "Write out the first few terms of the..."

avissidep

2021-02-21

Write out the first few terms of the series

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

What is the series' sum?

Velsenw

Skilled2021-02-22Added 91 answers

Consider the series

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

\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{5^{n}} = \frac{(- 1)^{0}}{5^{0}} + \frac{(- 1)^{1}}{5^{1}} + \frac{(- 1)^{2}}{5^{2}} + \frac{(- 1)^{3}}{5^{3}} + . . .

\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{5^{n}} = \frac{1}{1} + \frac{- 1}{5} + \frac{1}{5^{2}} + \frac{- 1}{5^{3}} + . . .

\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{5^{n}} = 1 - \frac{1}{5} + \frac{1}{5^{2}} - \frac{1}{5^{3}} + . . .

Series sum

\sum_{n = 0}^{\infty} \frac{(- 1)^{n}}{5^{n}} = 1 - \frac{1}{5} + \frac{1}{5^{2}} - \frac{1}{5^{3}} + . . .

It is a infinite geometric series with first term

a = 1

and common ratio

r = - \frac{1}{5}

Sum of a infinite geometric series is given by

S = \frac{a}{1 - r}

S = \frac{1}{1 - (- \frac{1}{5})}

S = \frac{1}{1 + \frac{1}{5}}

S = \frac{1}{\frac{5 + 1}{5}}

S = \frac{5}{6}

First term is

= 1

Sum

= \frac{5}{6}

Jeffrey Jordon

Expert2021-12-27Added 2605 answers

Answer is given below (on video)

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