Let’s analyze each series step by step to determine whether they converge, diverge, or conditionally/absolutely converge.

Problem 16: $\sum_{n=1}^\infty \arctan(n)$

• The term $\arctan(n)$ grows unbounded as $n \to \infty$, meaning it does not approach zero.
• A necessary condition for a series to converge is that its terms must approach zero.
• Since $\arctan(n) \to \frac{\pi}{2}$ (not zero) as $n \to \infty$, the series diverges.

Problem 17: $\sum_{n=1}^\infty \frac{\cos(n\pi)}{n}$

• $\cos(n\pi) = (-1)^n$, so the series becomes: $\sum_{n=1}^\infty \frac{(-1)^n}{n}.$
• This is the alternating harmonic series, which is known to converge conditionally.
• However, it does not converge absolutely because the harmonic series $\sum_{n=1}^\infty \frac{1}{n}$ diverges.

Result: This series converges conditionally.

Problem 18: $\sum_{n=1}^\infty \frac{(-1)^n \arcsin(\frac{1}{n})}{n^3}$

1. Absolute Convergence Test:
   • Consider the absolute value of the terms: $\left| \frac{(-1)^n \arcsin\left(\frac{1}{n}\right)}{n^3} \right| = \frac{\arcsin\left(\frac{1}{n}\right)}{n^3}.$
   • For large $n$, $\arcsin\left(\frac{1}{n}\right) \sim \frac{1}{n}$ (as $\arcsin(x) \sim x$ for small $x$).
   • The terms behave like $\frac{1/n}{n^3} = \frac{1}{n^4}$, and $\sum_{n=1}^\infty \frac{1}{n^4}$ converges (p-series with $p = 4 > 1$).
   • Thus, the series converges absolutely.

Result: This series converges absolutely.

Final Results:

1. Problem 16: Diverges.
2. Problem 17: Converges conditionally.
3. Problem 18: Converges absolutely.

Do you want more details about any of these steps?
Here are 5 follow-up questions to deepen your understanding:

1. What is the difference between conditional and absolute convergence?
2. Why does $\arctan(n)$ growing unbounded imply divergence for Problem 16?
3. How does the alternating series test apply to Problem 17?
4. What is a p-series, and why is it important for Problem 18?
5. How can $\arcsin(1/n) \sim 1/n$ be rigorously justified for large $n$?

Tip: Always check whether the terms of a series tend to zero before analyzing further properties!