#### To determine

**To find:** The values of *x* if the given series is convergent and obtain the sum of the series.

#### Answer

The interval of convergence is (−15,15) and the sum of the series is −5x1+5x.

#### Explanation

**Given:**

The series is ∑n=1∞(−5)nxn.

**Result used:**

The geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is converges if |r|<1 and its sum is a1−r, where *a* is the first term and *r* is the common ratio of the series.

**Calculation:**

Obtain the value of *x* (the interval of converges).

The given series can be expressed as follows,

∑n=1∞(−5)nxn=∑n=1∞(−5x)n=(−5x)+(−5x)2+(−5x)3+(−5x)4+⋯

Clearly, it is geometric series with first term of the series is a=−5x and common ratio is r=−5x.

Use the Result stated above, the geometric series ∑n=1∞(−5)nxn is converges if |r|<1.

|−5x|<15|x|<1|x|<15

That is, −15<x<15.

Thus, the series is converges if x∈(−15,15).

Therefore, the interval of convergence is (−15,15).

Obtain the sum of the geometric series.

Since a=−5x and r=−5x, the sum of the series is computed as follows.

∑n=1∞(−5)nxn=−5x1−(−5x)=−5x1+5x

Therefore, the sum of the series is −5x1+5x.