3. (20 points) Determine if the following series converge conditionally, converge absolutely, or diverge. Make sure you state which test you use to make your conclusion.
(a) $\sum_{k=1}^{\infty} \frac{3 k^{2}+6 k+8}{2 k^{7}+2 k^{5}+1}$

Step 1 ：We are given the series \(\sum_{k=1}^{\infty} \frac{3 k^{2}+6 k+8}{2 k^{7}+2 k^{5}+1}\). To determine if the series converges conditionally, converges absolutely, or diverges, we can use the Ratio Test.

Step 2 ：The Ratio Test states that if the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term of the series is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive.

Step 3 ：Applying the Ratio Test to our series, we find that the limit of the ratio is 1, which means the Ratio Test is inconclusive. We need to use another test to determine if the series converges or diverges.

Step 4 ：We can use the Comparison Test. The Comparison Test states that if 0 ≤ a_n ≤ b_n for all n and the series of b_n converges, then the series of a_n also converges. If the series of b_n diverges, then the series of a_n also diverges.

Step 5 ：We can compare our series to the series of 1/k^5. The limit of the ratio of our series to the comparison series is 3/2, which is greater than 1.

Step 6 ：This means that our series diverges by the Comparison Test.

Step 7 ：\(\boxed{\text{The series } \sum_{k=1}^{\infty} \frac{3 k^{2}+6 k+8}{2 k^{7}+2 k^{5}+1} \text{ diverges.}}\)