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

Answer 1

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.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Samantha Adkison

The series ( \sum_{n=1}^{\infty} \cos(n\pi) ) converges if the terms in the series approach zero as ( n ) approaches infinity. In this case, ( \cos(n\pi) ) alternates between -1 and 1 as ( n ) increases, so the series oscillates and does not converge. Therefore, the series diverges.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.