determine if the geometric series converges or diverges. if it converges, find its sum\n1 +…

determine if the geometric series converges or diverges. if it converges, find its sum\n1 + \\frac{7}{8}+\\left(\\frac{7}{8}\\right)^{2}+\\left(\\frac{7}{8}\\right)^{3}+\\cdots+\\left(\\frac{7}{8}\\right)^{n}+\\cdots\nselect the correct choice below and fill in any answer boxes within your choice\na. the sum is\n(type an integer or a simplified fraction )\nb. the series diverges

determine if the geometric series converges or diverges. if it converges, find its sum\n1 + \\frac{7}{8}+\\left(\\frac{7}{8}\\right)^{2}+\\left(\\frac{7}{8}\\right)^{3}+\\cdots+\\left(\\frac{7}{8}\\right)^{n}+\\cdots\nselect the correct choice below and fill in any answer boxes within your choice\na. the sum is\n(type an integer or a simplified fraction )\nb. the series diverges

Answer

Explanation:

Step1: Identify the first term (a) and common ratio (r)

For the geometric series (1+\frac{7}{8}+(\frac{7}{8})^{2}+(\frac{7}{8})^{3}+\cdots+(\frac{7}{8})^{n}+\cdots), the first term (a = 1) and the common ratio (r=\frac{7}{8}).

Step2: Check the convergence condition

A geometric series (\sum_{n = 0}^{\infty}a\cdot r^{n}) converges if (|r|\lt1). Since (|r|=\left|\frac{7}{8}\right|=\frac{7}{8}\lt1), the series converges.

Step3: Use the sum formula for a convergent geometric series

The sum formula for a convergent geometric series is (S=\frac{a}{1 - r}). Substitute (a = 1) and (r=\frac{7}{8}) into the formula: [ \begin{align*} S&=\frac{1}{1-\frac{7}{8}}\ &=\frac{1}{\frac{8 - 7}{8}}\ &=\frac{1}{\frac{1}{8}}\ &=8 \end{align*} ]

Answer:

A. The sum is (8)