How do you determine if the series the converges conditionally, absolutely or diverges given sum_(n=1)^oo (-1)^(n+1)arctan(n)?

Apr 13, 2018

The series diverges.

Explanation:

For ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} \arctan n$ we use the alternating series test:

The Alternating Series Test

An alternating series ${\sum}_{n = 1}^{\infty} {\left(- 1\right)}^{n + 1} {a}_{n}$ will converge iff:

• All the ${a}_{n}$ terms are positive (the sign of each term in the series alternates)
• The terms are eventually weakly decreasing (${a}_{n} \ge {a}_{n + 1}$ for large enough $n$)
• ${a}_{n} \to 0$

The series fulfils the first condition of the test but fails the second condition (and third) as $\arctan$ is an increasing function and ${\lim}_{n \to \infty} \arctan n = \pi \text{/} 2$.

And if the series doesn't converge conditionally then it doesn't converge absolutely.