S.O.S. Mathematics CyberBoard

[moonsenshi18]

Hi !
Just one more problem I need some help with!

How can I show that the summation n=1 to infinity of (-1)^n*n / 4^n is convergent by the Alternating Series Test?

I hope it is possible because I would like to estimate the sum by using the Alternating Series Estimation Theorem, which I can't use until I know that the given series is convergent by the A.S.T.

I m having trouble showing the limit as n approaches infinity of Bn (b sub-n) = 0.

Once again ideas by Monday would be wonderful!
I really do appreciate all the help so far

[Kungsman]

Do you know what the alternating series test (Leibniz' test) is? It's pretty straightforward to verify the conditions that has to be met...

Umm, by , do you perhaps mean



You should have learned long ago that exponentials grow much faster than power functions.

[qleak]

I'll use the AST as shown below:

http://en.wikipedia.org/wiki/Alternating_series_test

Your series is


So all you need to do is show that


The limit is calc I material unless, of course, you're asked to show that its zero.

[Kungsman]

One way to show that



is to use the binomial theorem. We have



so



and perhaps you can say something about what happens to this last fraction when x gets bigger.

[Kungsman]

[jinydu]

moonsenshi, don't you have anything more to say on the "estimating a p-series" thread?

[Marvin_M]

Don't forget that the alternating series test has a monotonicity condition as well.

Since only the tail determines convergence, we need to show that for some N,

[Kungsman]

[Marvin_M]

[Matt]

[Kungsman]