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

1 Answer
Apr 13, 2018

The series diverges.

Explanation:

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

The Alternating Series Test

An alternating series sum_(n=1)^oo(-1)^(n+1)a_nn=1(1)n+1an will converge iff:

  • All the a_nan terms are positive (the sign of each term in the series alternates)
  • The terms are eventually weakly decreasing (a_n>=a_(n+1)anan+1 for large enough nn)
  • a_n->0an0

The series fulfils the first condition of the test but fails the second condition (and third) as arctanarctan 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.