Evaluate the geometric series or state that it diverges.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
B. The series diverges.

Ask by Mitchell Gough. in the United States

Mar 30,2025

Question

Evaluate the geometric series or state that it diverges.

Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A.
B. The series diverges.

Ask by Mitchell Gough. in the United States

Mar 30,2025

UpStudy AI · Accepted Answer

We are given the geometric series

$$
\sum_{k=0}^{\infty}\left(\frac{1}{20}\right)^{k}.
$$

For a geometric series of the form

$$
\sum_{k=0}^{\infty} ar^k,
$$

the series converges if $$|r| < 1$$ and its sum is given by

$$
\frac{a}{1 - r}.
$$

In this case, the first term is $$a = 1$$ and the common ratio is

$$
r = \frac{1}{20}.
$$

Since

$$
\left|\frac{1}{20}\right| < 1,
$$

the series converges.

Now, we calculate the sum:

$$
\sum_{k=0}^{\infty}\left(\frac{1}{20}\right)^{k} = \frac{1}{1 - \frac{1}{20}}.
$$

Simplify the denominator:

$$
1 - \frac{1}{20} = \frac{20}{20} - \frac{1}{20} = \frac{19}{20}.
$$

Thus, the sum is

$$
\frac{1}{\frac{19}{20}} = \frac{20}{19}.
$$

Therefore, the correct choice is:

A. $$\sum_{k=0}^{\infty}\left(\frac{1}{20}\right)^{k}=\frac{20}{19}$$.
The sum of the series is $$ \frac{20}{19} $$.