infinite series (alternating series test)...need help
 

Part of testing for divergence/convergence involves the use of the "Alternating Series Test". I need to prove that a series is both DECREASING and the limit -> 0. I obviously know how to test the limit (which I usually do first) but determining if the series is decresing or not is harder to prove. The simple answer is to look at the denominator and if it "larger" than the numerator than the series is decreasing but my teacher says that is not a sufficient proof.
Please explain how to prove that a series is decreasing

thanks

Re: infinite series (alternating series test)...need help
 

Here's an example.

Show that sum (-1)^n n/(n^2+1) n = 1 to infinity converges.

Proof: This is an alternating series. Set a_n = n/(n^2+1). Then a_n --> 0.

Set f(x) = x/(x^2+1). Then f'(x) = (1-x^2)/(x^2+1)^2 which is < 0 for x > 1. Therefore f is a decreasing function and this shows that a_n decreases too.

Now apply the alternating series convergence theorem.

Re: infinite series (alternating series test)...need help
 

thank you

Re: infinite series (alternating series test)...need help
 

Usually simple algebra and mathematical induction will do it.

Using Jason's example

You want to compare

a(n) and a(n+1)

[n/(n^2+1)] and [(n+1)/((n+1)^2+1)]

cross multiply

n[((n+1)^2+1] and (n+1)(n^2+1)

n^3+2n^2+n+1 and n^3+n^2+n+1

n^2 and 0 which is obviously >

Go back and then a(n) > a(n+1) therefore decreasing

Re: infinite series (alternating series test)...need help
 

yah just take the derivative of the analagous function f(x)