Determine the sum of the following series.

$
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{5^{n}}
$

Question

Determine the sum of the following series.

$
\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{5^{n}}
$

StudyXY · Accepted Answer

: Recognize that this is an alternating series with terms in the form of $\frac{(-1)^{n-1}}{5^{n}}$. : To find the sum of an alternating series, we can use the alternating series test. : In this case, |a\_n| is simply $\frac{1}{5^{n}}$. : Now we can apply the formula from the alternating series test: : We can simplify this expression by combining the constants (-1)^(n+1) and $\frac{1}{5^{n}}$ into a single fraction: : To find the sum of this series, we can use the formula for the sum of an infinite geometric series, which is $\frac{a}{1-r}$, where a is the first term and r is the common ratio. : Plugging these values into the formula, we get: The sum of the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{5^{n}}$ is approximately 0.1667.

Q
QuestionAnatomy and Physiology

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Full Solution Locked

Step 1
: Recognize that this is an alternating series with terms in the form of \frac{(- 1)^{n- 1}}{5^{n}}.

Step 2
: To find the sum of an alternating series, we can use the alternating series test.

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