Convergence of Alternating Series: ln(1+x) <= x Hint for Absolute Convergence

[rcmango]

Homework Statement

Does this problem converge absolutely, conditionally, or does it diverge?

the equation: [/URL]

Homework Equations

also, the hint is to first show that ln(1 + x) <= x if x > 0

The Attempt at a Solution

It looks like an alternating series. not sure what the hint is implying or if its converging.

Thanks for any help.

 

[Gib Z]

Each term gets smaller and smaller, and converges to zero. It is absolutely convergent.

The ratio test will tell you it converges as well.

 

[HallsofIvy]

Each term gets smaller and smaller, and converges to zero. It is absolutely convergent.

Is that what you mean to say? Each term of
[tex]\Sigma_{n\rightarrow \infty}\frac{1}{n}[/itex]
"gets smaller and smaller, and converges to zero" but the series doesn't converge at all.

 

[mjsd]

when it is an alternating series you can use Leibniz test
your pic is not very clear... but my guess is that the hint is to help you establish one of the condition in the Leibniz test namely, the terms are getting smaller

Leibniz test:
If $$\sum_1^{\infty} (-1)^{n+1} b_n$$ such that all $$b_n>0$$ (ie alternating series) and $$b_{n+1} < b_n\; \forall\,n$$ and $$b_n\rightarrow 0$$, then series converges to S and $$|S-S_k|\leq b_{k+1}$$