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.