Let's analyze the problem and determine the appropriate convergence test for each series:

a)

$\sum_{n=1}^\infty \frac{4^n}{\cos(n) + 2}$

• The term $4^n$ grows exponentially, and this series does not resemble a geometric series or $p$-series. Instead, it is dominated by the numerator $4^n$, and the terms do not approach zero.
• Test to use: Divergence Test.

b)

$\sum_{n=1}^\infty \frac{n^4 - 12n - 8}{n + 3}$

• This series grows polynomially, and comparing the numerator and denominator for large $n$ reveals a behavior dominated by $n^4/n = n^3$. Neither geometric nor $p$-series fits this structure.
• Test to use: Divergence Test (because terms grow unbounded as $n \to \infty$).

c)

$\sum_{n=1}^\infty \frac{\sqrt[5]{n}}{n^6}$

• The general form is $\frac{n^{1/5}}{n^6} = n^{-29/5}$, which resembles a $p$-series with $p = 29/5 > 1$.
• Test to use: $p$-Series Test.

d)

$\sum_{n=1}^\infty \frac{4^n}{3^{2n}}$

• This can be rewritten as $\sum_{n=1}^\infty \left(\frac{4}{9}\right)^n$, which fits the structure of a geometric series with ratio $r = 4/9 < 1$.
• Test to use: Geometric Series Test.

Final Answers:

a) Divergence Test
b) Divergence Test
c) $p$-Series Test
d) Geometric Series Test

Would you like me to go into details about any of these tests or further explanations?

Relative Questions:

1. What is the difference between $p$-series and geometric series?
2. How can the divergence test be applied to identify non-converging series?
3. Why is $p > 1$ a critical condition for the $p$-series convergence?
4. How can a series be simplified to resemble a geometric series?
5. When does the divergence test fail to determine convergence?

Tip: Always simplify the series as much as possible before choosing the convergence test!