How do you determine whether the infinite sequence #a_n=ncos(npi)# converges or diverges?

Question

5000 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2014-09-24T11:08:15

The sequence #a_n=ncos(n pi)# diverges.

Let us look at some details.

Let us take the limit of its subsequence of only even terms.

#\lim_{n to infty}a_{2n}=lim_{n to infty}(2n)cos(2npi)#

since #cos(2npi)=1# for all integer #n#,

#=lim_{n to infty}(2n)=2(infty)=infty#

Since the subsequence diverges, the original sequence also diverges.

How do you determine whether the infinite sequence #a_n=n*cos(n*pi)# converges or diverges?