Question

question
which of the following gives the correct formula for evaluating the sum below?
5 - \frac{5}{3}+\frac{5}{9}+...
select the correct answer below:
\frac{5}{1 - \frac{1}{3}}
\frac{5}{1+\frac{5}{3}}
\frac{-\frac{1}{3}}{1 - 5}
\frac{-\frac{1}{3}}{1 + 5}
\frac{5}{1+\frac{1}{3}}

Explanation:

Step1: Identify the geometric - series

The given series $5-\frac{5}{3}+\frac{5}{9}+\cdots$ is a geometric series with first - term $a = 5$ and common ratio $r=-\frac{1}{3}$.

Step2: Recall the formula for the sum of an infinite geometric series

The formula for the sum of an infinite geometric series is $S=\frac{a}{1 - r}$ when $|r|\lt1$.

Step3: Substitute the values of $a$ and $r$ into the formula

Substituting $a = 5$ and $r=-\frac{1}{3}$ into the formula $S=\frac{a}{1 - r}$, we get $S=\frac{5}{1-(-\frac{1}{3})}=\frac{5}{1 + \frac{1}{3}}$.

Answer:

$\frac{5}{1+\frac{1}{3}}$