write out the first few terms of the series (sum_{n = 0}^{infty}\frac{(-1)^{n}}{8^{n}}). what is the series…

write out the first few terms of the series (sum_{n = 0}^{infty}\frac{(-1)^{n}}{8^{n}}). what is the series sum?\nthe first term is 1 (type an integer or a simplified fraction.)\nthe second term is (-\frac{1}{8}) (type an integer or a simplified fraction.)\nthe third term is (\frac{1}{64}) (type an integer or a simplified fraction.)\nthe fourth term is (-\frac{1}{512}) (type an integer or a simplified fraction.)\nif the series converges, what is the series sum? select the correct choice below and fill in any answer boxes within your choice\na the series converges. the series sum is (type an integer or a simplified fraction.)\nb the series diverges

write out the first few terms of the series (sum_{n = 0}^{infty}\frac{(-1)^{n}}{8^{n}}). what is the series sum?\nthe first term is 1 (type an integer or a simplified fraction.)\nthe second term is (-\frac{1}{8}) (type an integer or a simplified fraction.)\nthe third term is (\frac{1}{64}) (type an integer or a simplified fraction.)\nthe fourth term is (-\frac{1}{512}) (type an integer or a simplified fraction.)\nif the series converges, what is the series sum? select the correct choice below and fill in any answer boxes within your choice\na the series converges. the series sum is (type an integer or a simplified fraction.)\nb the series diverges

Answer

Explanation:

Step1: Identify the series type

This is a geometric series of the form (\sum_{n = 0}^{\infty}ar^{n}), where (a = 1) (when (n = 0), (a=\frac{(- 1)^{0}}{8^{0}}=1)) and (r=-\frac{1}{8}).

Step2: Apply the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is (S=\frac{a}{1 - r}), provided (|r|\lt1). Here, (|r|=\left|-\frac{1}{8}\right|=\frac{1}{8}\lt1).

Substitute (a = 1) and (r=-\frac{1}{8}) into the formula: [ \begin{align*} S&=\frac{1}{1-\left(-\frac{1}{8}\right)}\ &=\frac{1}{1+\frac{1}{8}}\ &=\frac{1}{\frac{8 + 1}{8}}\ &=\frac{8}{9} \end{align*} ]

Answer:

A. The series converges. The series' sum is (\frac{8}{9})