Question

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

Answer 1

The series diverges.

Explanation:

For `sum_(n=1)^oo(-1)^(n+1) arctan n` we use the alternating series test:

The Alternating Series Test

An alternating series `sum_(n=1)^oo(-1)^(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>=a_(n+1)` for large enough `n`)
• `a_n->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->oo)arctan n =pi"/"2`.

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