Answer:
To determine whether the geometric series [infinity] (2^n)(6n+1), n=0, is convergent or divergent, we can use the ratio test.
The ratio test states that if the limit of the absolute value of the ratio of the (n+1)th term to the nth term is less than 1, then the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive.
Let's apply the ratio test to this series:
|(2^(n+1))(6(n+1)+1)|
--------------------- = |2 * 6n + 13|
|(2^n)(6n+1)|
As n approaches infinity, the absolute value of the ratio simplifies to:
|2 * 6n + 13|
-------------
|2^n|
Dividing both the numerator and denominator by 2^n, we get:
|6n/2^n + 13/2^n|
------------------
1
As n approaches infinity, 6n/2^n approaches 0, and 13/2^n approaches 0. Therefore, the limit of the absolute value of the ratio is 0 + 0 = 0, which is less than 1.
Since the limit of the absolute value of the ratio is less than 1, the series [infinity] (2^n)(6n+1), n=0, is convergent.