Does the infinite geometric series converge or diverge? 1 + 10 + 100 + 1,000 + Choices: (A)converge (B)diverge

Find Common Ratio: To determine whether the infinite geometric series converges or diverges, we need to find the common ratio (r) of the series. The common ratio is the factor by which each term is multiplied to get the next term. In this series, we can find the common ratio by dividing the second term by the first term, the third term by the second term, and so on. Let's calculate the common ratio using the first two terms: r = 10 / 1 = 10Calculate Sum Formula: Now that we have the common ratio, we can use the formula for the sum of an infinite geometric series, which is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio. This formula only applies if the absolute value of the common ratio is less than 1 (|r| < 1). Since our common ratio is 10, which is greater than 1, the series does not meet the criteria for convergence.Series Diverges: Because the common ratio is greater than 1, the terms of the series will continue to grow larger without bound. Therefore, the series does not have a finite sum and diverges.

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Does the infinite geometric series converge or diverge? $\newline$ $1 + 10 + 100 + 1,000 + \dots$ $\newline$ Choices: $\newline$ (A) converge $\newline$ (B) diverge

View step-by-step help

Home

Math Problems

Precalculus

Convergent and divergent geometric series

Full solution

Q. Does the infinite geometric series converge or diverge?

\newline

1 + 10 + 100 + 1,000 + \dots

\newline

Choices:

\newline

(A) converge

\newline

(B) diverge

Find Common Ratio: To determine whether the infinite geometric series converges or diverges, we need to find the common ratio $r$ of the series. $\newline$ The common ratio is the factor by which each term is multiplied to get the next term. $\newline$ In this series, we can find the common ratio by dividing the second term by the first term, the third term by the second term, and so on. $\newline$ Let's calculate the common ratio using the first two terms: $\newline$ $r = \frac{10}{1} = 10$
Calculate Sum Formula: Now that we have the common ratio, we can use the formula for the sum of an infinite geometric series, which is $S = \frac{a}{(1 - r)}$ , where $S$ is the sum, $a$ is the first term, and $r$ is the common ratio. $\newline$ This formula only applies if the absolute value of the common ratio is less than $1$ (|r| < 1). $\newline$ Since our common ratio is $10$ , which is greater than $1$ , the series does not meet the criteria for convergence.
Series Diverges: Because the common ratio is greater than $1$ , the terms of the series will continue to grow larger without bound. $\newline$ Therefore, the series does not have a finite sum and diverges.