Unravelling the Mystery of Cn: How to Find Convergent Sequences

[BuBbLeS01]

Sequences HELP!

Homework Statement

Show that the sequence Cn = [(-1)^n * 1/n!]

Homework Equations

The Attempt at a Solution

This is an example in my book but I am not understanding it...

It says to find 2 convergent sequences that can be related to the given sequence. 2 possibilities are An = -1/2^n and Bn = 1/2^n
.....
where are they getting this from??

 

[RyanSchw]

When trying to determine if a sequence converges it needs to approach a certain number. By pluging in numbers for the value of n you'll notice that

Cn = [(-1)^n * 1/n!]

will jump back and fourth between positive and negative values, and therefore isn't approaching a single value. You then need to find a graph both above and below that sequence that converge, you do this by the squeeze theorem.

If you graph all three of those sequences you'll notice that Cn = [(-1)^n * 1/n!] lies in-between. You can therefore say that because

1/2^n

and

-1/2^n

converge, the value in-between them converges also. What would that be the?

 

[BuBbLeS01]

but how do you come up with 1/2^n and -1/2^n?? The value would be 0?

 

[RyanSchw]

Yes all the values converge to zero. I really don't like to use the squeeze theorem, whenever I have to use it it's just by trial and error.

There should however be another theorem which relates to sequences that jump back and fourth. It's called the Absolute Value Theorem, which would allow you to more easily find the convergence of a sequence like this.

 

[BuBbLeS01]

ok thank you so much!