Answer:
To determine whether the series converges, we can use the ratio test and find the limit of the absolute value of the ratio of the (n+1)th term to the nth term as n approaches infinity. If the limit is less than 1, the series converges. If it is greater than 1, the series diverges. If it is equal to 1, the test is inconclusive. After performing the calculations, we find that the series converges with a sum of -9/2.
Step-by-step explanation:
To determine whether the series ∑k=1∞(2k+4) / (3k+1) converges and if so, find its sum, 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 as n approaches infinity is less than 1, then the series converges. If the limit is greater than 1, the series diverges.
If the limit is equal to 1, the test is inconclusive. Let's apply this test to the given series:
- Calculate the limit of the absolute value of ((2(n+1)+4) / (3(n+1)+1)) / ((2n+4) / (3n+1)) as n approaches infinity.
- If the limit is less than 1, then the series converges. If it is greater than 1, the series diverges. If it is equal to 1, the test is inconclusive.
After performing the calculations, we find that the limit is equal to 4/3, which is less than 1. Therefore, the series converges.
To find the sum of the series, we need to evaluate the sum formula.
In this case, the sum formula is S = (a / (1 - r)), where 'a' is the first term and 'r' is the common ratio.
In the given series, the first term is (2(1)+4) / (3(1)+1) = 6 / 4 = 3/2 and the common ratio is ((2(n+1)+4) / (3(n+1)+1)) / ((2n+4) / (3n+1)), which we found to be 4/3.
Substituting these values into the sum formula, we get S = ((3/2) / (1 - 4/3)), which simplifies to S = ((3/2) / (-1/3)), or S = -9/2.