To determine whether the series

$\sum_{n=1}^{\infty} \frac{(-1)^n \arctan(n)}{n^2}$

is absolutely convergent, conditionally convergent, or divergent, let's analyze it step-by-step.

Step 1: Check for Absolute Convergence

To test for absolute convergence, consider the absolute value of the series terms:

$\sum_{n=1}^{\infty} \left| \frac{(-1)^n \arctan(n)}{n^2} \right| = \sum_{n=1}^{\infty} \frac{|\arctan(n)|}{n^2}.$

Since $\arctan(n)$ is bounded for all $n$, we know that $|\arctan(n)|$ grows very slowly and is less than or equal to $\frac{\pi}{2}$ as $n \to \infty$. So, the series we are considering for absolute convergence is comparable to

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

which is a convergent $p$-series with $p = 2 > 1$.

Therefore, $\sum_{n=1}^{\infty} \frac{|\arctan(n)|}{n^2}$ converges, implying that the original series is absolutely convergent.

Conclusion

Since the series is absolutely convergent, it is also convergent.

Would you like further details on this analysis or have any specific questions?

Here are five related questions to deepen your understanding:

1. How does the alternating series test work for determining conditional convergence?
2. What are the properties of the arctan function that influence convergence in series?
3. How does the $p$-series test determine convergence, and why is $p = 2$ significant here?
4. What are the differences between absolute and conditional convergence?
5. How would the convergence result change if the $n^2$ in the denominator were replaced with $n$?

Tip: When testing convergence, start by considering absolute convergence, as it is generally easier to establish than conditional convergence.