To determine

Whether the geometric series is convergent or divergent and obtain the sum if the series is convergent.

Answer

The series is divergent.

Explanation

Given:

The series is 3−4+163−649+⋯ .

Here, a=3 and r=−43.

Result used:

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

Calculation:

The first term of the series is a=3 .

The common ratio of the series is,

r=Second termFirst term=−43

The absolute value of r is,

|r|=|−43|=43=1.33>1

Since |r|>1 and by using the Result stated above, the series is divergent.

Therefore, the series is divergent.