Find the sum of the infinite geometric series. -1 - 1/4 - 1/16 - 1/64 + Write your answer as an integer or a fraction in simplest form. ______

Identify Terms: Identify the first term (a) and the common ratio (r) of the geometric series. The first term a is -1, and the common ratio r can be found by dividing the second term by the first term, which is (-1/4) / (-1) = 1/4.Check Common Ratio: Check if the common ratio's absolute value is less than 1 to ensure the series converges. The common ratio r is 1/4, and its absolute value |1/4| is less than 1, which means the series converges.Use Sum Formula: 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. Substitute the values of a and r into the formula to find the sum. S = (-1) / (1 - (1/4))Simplify Expression: Simplify the expression to find the sum. S = (-1) / (1 - 1/4) S = (-1) / (3/4) S = (-1) * (4/3) S = -4/3

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Find the sum of the infinite geometric series. $\newline$ $-1 - \frac{1}{4} - \frac{1}{16} - \frac{1}{64} +$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ $\_\_$

View step-by-step help

Home

Math Problems

Precalculus

Find the value of an infinite geometric series

Full solution

Q. Find the sum of the infinite geometric series.

\newline

-1 - \frac{1}{4} - \frac{1}{16} - \frac{1}{64} +

\newline

Write your answer as an integer or a fraction in simplest form.

\newline

\_\_

Identify Terms: Identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a$ is $-1$ , and the common ratio $r$ can be found by dividing the second term by the first term, which is $(-1/4) / (-1) = 1/4$ .
Check Common Ratio: Check if the common ratio's absolute value is less than $1$ to ensure the series converges. $\newline$ The common ratio $r$ is $\frac{1}{4}$ , and its absolute value $|\frac{1}{4}|$ is less than $1$ , which means the series converges.
Use Sum Formula: 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. $\newline$ Substitute the values of $a$ and $r$ into the formula to find the sum. $\newline$ $S = \frac{-1}{1 - (\frac{1}{4})}$
Simplify Expression: Simplify the expression to find the sum. $\newline$ $S = \frac{-1}{1 - \frac{1}{4}}$ $\newline$ $S = \frac{-1}{\frac{3}{4}}$ $\newline$ $S = (-1) \times \frac{4}{3}$ $\newline$ $S = -\frac{4}{3}$