• Jul 11th 2009, 07:32 PM
Hikari
Is it okay to use Comparison Theorem to prove that a sequence converges? I would think so, but I recall my professor commenting on that in class by saying that it works with series. I don't know whether or not he meant that to exclude sequences.

If not, why doesn't it work? I appreciate the help!
• Jul 11th 2009, 10:07 PM
Jhevon
Quote:

Originally Posted by Hikari
Is it okay to use Comparison Theorem to prove that a sequence converges? I would think so, but I recall my professor commenting on that in class by saying that it works with series. I don't know whether or not he meant that to exclude sequences.

If not, why doesn't it work? I appreciate the help!

The comparison test works with series, not sequences.

to see that it does not work in general for sequences, consider the following two sequences

\$\displaystyle \{ a_n \} = 2,2,2,2,2 ...\$ and

\$\displaystyle \{ b_n \} = -1,1,-1,1,...\$

note that \$\displaystyle |b_n| \le |a_n|\$ for all \$\displaystyle n\$ and \$\displaystyle \{ a_n \}\$ converges. Yet, \$\displaystyle \{ b_n \}\$ does not converge
• Jul 12th 2009, 11:26 AM
Hikari
Thanks so much!