Given the series $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ answer the following questions. Answer Attempt 1 out of 2 $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} \underbrace{ ext { is }} $$ a geometric-series with the common ratio $$ r=\square $$ series will

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Given the series $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} $$ answer the following questions. Answer Attempt 1 out of 2 $$ \sum_{n=0}^{\infty} \frac{1}{(-5)^{2 n+1}} \underbrace{	ext { is }} $$ a geometric-series with the common ratio $$ r=\square $$ series will

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We start with the given series:

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

**Step 1. Rewrite the term**

Notice that the exponent $$2n+1$$ can be split as:

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

Since $$(-5)^{2n} = (25)^n$$ (because $$(-5)^2=25$$), we can rewrite the denominator:

$$
(-5)^{2n+1} = (-5) \cdot (25)^n 
$$

So each term of the series becomes:

$$
\frac{1}{(-5)^{2n+1}} = \frac{1}{-5 \cdot 25^n} = -\frac{1}{5} \cdot \frac{1}{25^n}
$$

**Step 2. Factor out the constant term**

The series now becomes:

$$
-\frac{1}{5} \sum_{n=0}^{\infty} \frac{1}{25^n}
$$

**Step 3. Identify the geometric series**

Recall that a geometric series in the form

$$
\sum_{n=0}^{\infty} ar^n
$$

has first term $$a$$ and common ratio $$r$$. For our series, the summation part is:

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

Thus, we have:

$$
a = 1 \quad \text{and} \quad r = \frac{1}{25}
$$

**Step 4. Answer the question**

The series $$\sum_{n=0}^{\infty} \frac{1}{(-5)^{2n+1}}$$ is a geometric series with the common ratio

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

Since $$\left|\frac{1}{25}\right| < 1$$, the series converges.
The series is a geometric series with a common ratio of $$ \frac{1}{25} $$.

Given the series answer the following questions.
Answer Attempt 1 out of 2
a geometric-series with the common
ratio
series will

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Given the series answer the following questions. Answer Attempt 1 out of 2 a geometric-series with the common ratio series will