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

1 Answer
Apr 13, 2018

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.