#### 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 (194,214) and the sum of the series is 14x−19

#### Explanation

**Given:**

The series is ∑n=0∞(−4)n(x−5)n.

**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=0∞(−4)n(x−5)n=∑n=0∞((−4)(x−5))n=∑n=0∞(−4(x−5))n=(−4(x−5))0+(−4(x−5))1+(−4(x−5))2+(−4(x−5))3+⋯=1+(−4(x−5))+(−4(x−5))2+(−4(x−5))3+⋯

Clearly, it is geometric series with first term of the series is a=1 and common ratio is r=−4(x−5).

Use the Result stated above, the geometric series ∑n=0∞(−4)n(x−5)n is converges if |r|<1.

|−4(x−5)|<14|x−5|<1|x−5|<14−14<x−5<14

Add both sides of the inequality with 5.

−14+5<x−5+5<14+5−1+204<x<1+204194<x<214

Thus, the series is converges if x∈(194,214) .

Therefore, the interval of convergence is (194,214) .

Obtain the sum of the geometric series.

Since a=1 and r=−4(x−5), the sum of the series is computed as follows.

∑n=0∞(−4)n(x−5)n=11−(−4(x−5))=11+4(x−5)=11+4x−20=14x−19

Therefore, the sum of the series is 14x−19.