Expert: Paul Klarreich - 4/29/2009

Question
If you have an oscillating function that is approaching 0 at infinity but it jumping back and forth from positive to negative because of a negative base to the nth power, does a limit exist.  Ex: (-3)^n * n/(n^2 +1)


Answer
Questioner:   Sandi
Country:  United States
Category:  Calculus
Private:  No
 
Subject:  limits of oscillating series
Question:  If you have an oscillating function that is approaching 0 at infinity but is jumping back and forth from positive to negative because of a negative base to the nth power, does a limit exist.  Ex: (-3)^n * n/(n^2 +1)
.........................
Hi, Sandi,

Your function is inconsistent with your question.
                 n
lim   [(-3)^n  ------- ]
n->inf         n^2 + 1

is not zero.  (-3)^n approaches infinity much faster than the second factor approaches zero.

But the answer is yes.  If  lim S[n] = 0, then the limit exists.

For example:

                    n
lim   [cos(n pi)  ------- ] = 0
n->inf            n^2 + 1

because  cos (n pi) = +- 1, depending...

Does this help?

BTW, a series and a sequence are not the same thing.  Watch your use of vocabulary.  Sloppy use of the vocabulary of math is a good way to fail analysis.