Let’s analyze each series and identify the appropriate convergence test without performing the test itself.

Series Analysis and Test Selection

a) $\sum_{n=1}^{\infty} \frac{\sqrt[7]{n}}{4 - n^4}$

The general term here has a $n^4$ term in the denominator, which will dominate for large $n$. This suggests the term resembles a $p$-series or could be compared to one. Therefore:

• Suggested Test: Limit Comparison Test or Comparison Test, comparing to $\frac{1}{n^4}$.

b) $\sum_{n=1}^{\infty} \frac{-115^n}{(n - 4)!}$

This series has factorial terms in the denominator, which grow faster than any exponential or polynomial term in the numerator. Series with factorials are well-suited for:

• Suggested Test: Ratio Test, since factorials in the terms make it effective for this test.

c) $\sum_{n=1}^{\infty} \left(\frac{n - 12n}{n^3 - 4}\right)^n$

This series contains a term raised to the $n$-th power. Series in this form typically suggest the Root Test is ideal, since it allows us to simplify the $n$-th power.

• Suggested Test: Root Test.

d) $\sum_{n=1}^{\infty} \frac{(-1)^n n^5}{4e^{2n}}$

This is an alternating series with an exponential growth factor in the denominator. The exponential term suggests rapid decay, so:

• Suggested Test: Ratio Test, because it is effective for handling terms with exponential factors.

Summary of Tests for Each Series

1. (a) Limit Comparison Test or Comparison Test
2. (b) Ratio Test
3. (c) Root Test
4. (d) Ratio Test

Would you like more details on any of these tests, or have any questions about specific series?

Here are some related questions to deepen understanding:

1. How do you decide between using the Comparison Test and the Limit Comparison Test?
2. What makes the Ratio Test particularly suitable for series involving factorials or exponentials?
3. In what cases is the Root Test generally preferred over other tests?
4. Why is the Alternating Series Test not suggested for series (d), even though it is alternating?
5. How do the terms $p$-series or geometric series assist in making comparisons in convergence tests?

Tip: When a series has terms that alternate in sign, always consider whether an alternating series test might apply, especially if the terms decrease in absolute value.