Question

Use an appropriate test to determine whether the following series converges. ∑ k=1[infinity]e 6k

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Select the correct choice below and fill in the answer box to complete your choice. (Type an exact answer.)

Answer 1

The series diverges because e^6 is greater than 1.

Step-by-step explanation:

To determine if the series ∑ k=1∞ e^(6k) converges, we can use the ratio test.

The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges. If the limit is greater than 1, the series diverges. And if the limit is equal to 1, the test is inconclusive.

Let's apply the ratio test to this series:

lim |(e^(6(k+1))) / (e^(6k))| as k approaches infinity

= lim |e^(6k+6) / e^(6k)| as k approaches infinity

= lim |e^(6k) * e^6 / e^(6k)| as k approaches infinity

= lim |e^6| as k approaches infinity

= e^6

Since e^6 is greater than 1, the limit of the ratio is greater than 1, which means the series diverges.

Therefore, the series ∑ k=1∞ e^(6k) diverges.