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Find the sum of the infinite geometric series. $\newline$ $1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \dots$ $\newline$ Write your answer as an integer or a fraction in simplest form. $\newline$ ______

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Find the value of an infinite geometric series

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Q. Find the sum of the infinite geometric series.

\newline

1 + \frac{2}{3} + \frac{4}{9} + \frac{8}{27} + \dots

\newline

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

\newline

______

Identify first term and common ratio: First, we need to identify the first term $a$ and the common ratio $r$ of the geometric series. $\newline$ The first term $a$ is $1$ . $\newline$ The common ratio $r$ is the factor by which each term is multiplied to get the next term. In this case, each term is multiplied by $\frac{2}{3}$ to get the next term. $\newline$ So, $r = \frac{2}{3}$ .
Use formula for infinite series sum: To find the sum of an infinite geometric series, we use the formula $S = \frac{a}{1 - r}$ , where $S$ is the sum of the series, $a$ is the first term, and $r$ is the common ratio. This formula only works if the absolute value of $r$ is less than $1$ , which is true in this case since $|\frac{2}{3}| < 1$ .
Plug in values for calculation: Now we can plug the values of $a$ and $r$ into the formula to find the sum of the series. $S = \frac{1}{1 - \frac{2}{3}}$
Calculate denominator of fraction: We calculate the denominator of the fraction: $1 - \frac{2}{3} = \frac{3}{3} - \frac{2}{3} = \frac{1}{3}$
Find sum by dividing first term: Now we can find the sum $S$ by dividing the first term by the result we just found: $S = \frac{1}{(1/3)}$
Multiply to get final sum: To divide by a fraction, we multiply by its reciprocal. So we multiply $1$ by the reciprocal of $\frac{1}{3}$ , which is $\frac{3}{1}$ or just $3$ . $\newline$ $S = 1 \times 3 = 3$

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