How do you determine the convergence or divergence of #sum_(n=1)^(oo) cosnpi#?

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Answer 1 · 2017-02-07T09:39:03

The series:

#sum_(n=1)^oo cos n pi #

is indeterminate

A necessary condition for a series to be convergent is that the sequence of its terms is infinitesimal, that is #lim_(n->oo) a_n =0#

We have:

#cosnpi = (-1)^n#

so the series is not convergent.

If we look at the partial sums we have:

#s_1 = -1#

#s_2 = -1+1=0#

#s_3 = 0-1=-1#

#...#

Clearly the partial sums oscillate, with all the sums of even order equal to zero and all the sums of odd order equal to #-1#.

Thus, there is no limit for #{s_n}# and the series is indeterminate.