To determine whether the given series

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

converges absolutely, conditionally, or diverges, let's analyze the behavior of the terms and the series:

Step 1: Behavior of $\arctan(n)$

The $\arctan(n)$ function approaches $\pi/2$ as $n \to \infty$, and its value grows without bound. Consequently, the terms $\arctan(n)$ do not approach 0 as $n \to \infty$.

Step 2: Divergence Test

A necessary condition for the convergence of any infinite series $\sum a_n$ is that $\lim_{n \to \infty} a_n = 0$. In this case:

$\lim_{n \to \infty} \arctan(n) \neq 0.$

Since the terms do not approach 0, the series diverges.

Conclusion

The series $\sum_{n=1}^\infty \arctan(n)$ diverges. The correct answer is:

• Diverges

Would you like further explanation or clarification?

Here are five related questions to expand your understanding:

1. What is the divergence test, and when should it be applied?
2. How does the behavior of $\arctan(n)$ compare to other functions like $\ln(n)$ or $1/n$?
3. What is the difference between absolute and conditional convergence?
4. How can you prove convergence or divergence for a series involving $\arctan$ with different transformations (e.g., $\arctan(1/n)$)?
5. What happens if the terms of the series involve $\arctan(n)/n^2$?

Tip: Always test whether the terms of a series approach zero before considering any convergence tests!