Find the sum of the infinite geometric series. -1 - 1/3 - 1/9 - 1/27 + Write your answer as an integer or a fraction in simplest form. ______

Identify Terms and Ratio: To find the sum of an infinite geometric series, we need to identify the first term (a) and the common ratio (r) of the series. The formula for the sum of an infinite geometric series is S = a / (1 - r), where |r| < 1. In the given series, the first term is -1 and the common ratio is the ratio between any two consecutive terms, which is 1/3 divided by 1 or -1/3 divided by -1, which gives us 1/3.Apply Sum Formula: Now we apply the formula for the sum of an infinite geometric series: S = a / (1 - r). Here, a = -1 and r = 1/3. So, S = -1 / (1 - 1/3).Calculate Denominator: We calculate the denominator of the fraction: 1 - 1/3 = 3/3 - 1/3 = 2/3.Substitute Denominator: Now we substitute the denominator back into the formula: S = -1 / (2/3).Multiply by Reciprocal: To divide by a fraction, we multiply by its reciprocal. So, S = -1 * (3/2).Perform Multiplication: We perform the multiplication: S = -3/2. This is the sum of the infinite geometric series in simplest fractional form.

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. $\newline$ $-1 - \frac{1}{3} - \frac{1}{9} - \frac{1}{27} +$ $\newline$ 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}{3} - \frac{1}{9} - \frac{1}{27} +

\newline

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

\newline

______

Identify Terms and Ratio: To find the sum of an infinite geometric series, we need to identify the first term $a$ and the common ratio $r$ of the series. The formula for the sum of an infinite geometric series is $S = \frac{a}{1 - r}$ , where |r| < 1. In the given series, the first term is $-1$ and the common ratio is the ratio between any two consecutive terms, which is $\frac{1}{3}$ divided by $1$ or $\frac{-1}{3}$ divided by $-1$ , which gives us $\frac{1}{3}$ .
Apply Sum Formula: Now we apply the formula for the sum of an infinite geometric series: $S = \frac{a}{(1 - r)}$ . Here, $a = -1$ and $r = \frac{1}{3}$ . So, $S = \frac{-1}{(1 - \frac{1}{3})}$ .
Calculate Denominator: We calculate the denominator of the fraction: $1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3}$ .
Substitute Denominator: Now we substitute the denominator back into the formula: $S = -1 / (\frac{2}{3})$ .
Multiply by Reciprocal: To divide by a fraction, we multiply by its reciprocal. So, $S = -1 \times \left(\frac{3}{2}\right)$ .
Perform Multiplication: We perform the multiplication: $S = -\frac{3}{2}$ . This is the sum of the infinite geometric series in simplest fractional form.