Given the series $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ answer the following questions. Answer Attempt 1 out of 2 $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ is $$ \vee $$ a geometric-series with the common ratio $$ r=\frac{1}{25} \cdot \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ converges $$ \vee $$ since a geometric series will converge if and only if $$ |\mathrm{r}|

Question

Given the series $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ answer the following questions. Answer Attempt 1 out of 2 $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ is $$ \vee $$ a geometric-series with the common ratio $$ r=\frac{1}{25} \cdot \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ converges $$ \vee $$ since a geometric series will converge if and only if $$ |\mathrm{r}|<1 \vee $$. The sum of the series is $$ \square $$

UpStudy AI · Accepted Answer

We start with the series

$$
\sum_{n=0}^{\infty} \frac{1}{(-5)^{2n+1}}.
$$

1. Write the general term as

$$
\frac{1}{(-5)^{2n+1}} = \frac{1}{(-5) \cdot (-5)^{2n}}.
$$

2. Notice that

$$
(-5)^{2n} = \left((-5)^2\right)^n = 25^n.
$$

Thus, the series can be rewritten as

$$
\sum_{n=0}^{\infty} \frac{1}{(-5)} \cdot \frac{1}{25^n} = \frac{1}{-5} \sum_{n=0}^{\infty} \left(\frac{1}{25}\right)^n.
$$

3. The inner sum

$$
\sum_{n=0}^{\infty} \left(\frac{1}{25}\right)^n
$$

is a geometric series with first term $$ a = 1 $$ and common ratio

$$
r = \frac{1}{25}.
$$

Since $$|r| = \frac{1}{25} < 1$$, the series converges.

4. The sum of a convergent geometric series is given by

$$
\sum_{n=0}^{\infty} r^n = \frac{1}{1 - r}.
$$

Thus, we have

$$
\sum_{n=0}^{\infty} \left(\frac{1}{25}\right)^n = \frac{1}{1 - \frac{1}{25}} = \frac{1}{\frac{24}{25}} = \frac{25}{24}.
$$

5. Now, include the factor $$\frac{1}{-5}$$:

$$
\sum_{n=0}^{\infty} \frac{1}{(-5)^{2n+1}} = \frac{1}{-5} \cdot \frac{25}{24} = -\frac{25}{120} = -\frac{5}{24}.
$$

Thus, the series converges and its sum is

$$
-\frac{5}{24}.
$$
The series $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2n+1}} $$ is a geometric series with a common ratio $$ r = \frac{1}{25} $$. Since $$ |r| < 1 $$, the series converges. The sum of the series is $$ -\frac{5}{24} $$.

Given the series answer the following questions.
Answer Attempt 1 out of 2
is a geometric-series with the common
ratio converges since a geometric
series will converge if and only if . The sum of the series
is

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Given the series answer the following questions. Answer Attempt 1 out of 2 is a geometric-series with the common ratio converges since a geometric series will converge if and only if . The sum of the series is