Convergence and Comparison: Analyzing the Alternating Series Test

[Bazzinga]

So I have this series:

$$\sum^{infinity}_{n=3}(-1)^{n-1}\frac{ln(n)}{n}$$

And I'm trying to use the AST to find out if it converges or not.
First of all, I'm stuck trying to show that ln(n)/n is decreasing...

But then after that. I'm assuming I can compare it with 1/n to show that it diverges absolutely, but converges conditionally (since the limit as n -> infinity of ln(n)/n is 0)

I was just wondering what converging absolutely and conditionally meant? We learned in class that absolute convergence implies convergence, does this mean that if its only conditionally convergent it doesn't converge?

 

[micromass]

So I have this series:

$$\sum^{infinity}_{n=3}(-1)^{n-1}\frac{ln(n)}{n}$$

And I'm trying to use the AST to find out if it converges or not.
First of all, I'm stuck trying to show that ln(n)/n is decreasing...

Consider the continuous extension of this function, i.e. consider the function

$$f:]0,+\infty[\rightarrow \mathbb{R}:x\rightarrow \frac{ln(x)}{x}$$

it suffices to show that this function is decreasing (from a certain point on). To show this, it suffices to calculate the derivative of the function and seeing where it is negative.

But then after that. I'm assuming I can compare it with 1/n to show that it diverges absolutely, but converges conditionally (since the limit as n -> infinity of ln(n)/n is 0)

I was just wondering what converging absolutely and conditionally meant? We learned in class that absolute convergence implies convergence, does this mean that if its only conditionally convergent it doesn't converge?

Well, absolute convergence happens when the series of absolute values converge. Absolute convergence is stronger then convergence. However, there are series which converge and do not converge absolutely. This type of convergence is conditionally convergence.An example of this phenomenon is

$$\sum\frac{(-1}^n}{n}$$

So series can do three kind of things: they can diverge, they can converge absolutely and they can converge conditionally.

 

[Bazzinga]

Ok, I understand that, but what's the difference between conditional convergence and absolute convergence? I always thought it was just 1s and 0s, it either converges or not, but conditional convergence is in between? Does is converge slower or something?

 

[micromass]

No, there is nothing slower about conditional convergence (that I know of). Both conditional and absolute convergence is convergence. The only difference is that conditional convergent series do not converge absolute, while absolutely convergent sequences do.

The reason that mathematicians make the distinction is because there are a lot of nice properties of absolute convergent series that conditional convergent series do not have. For instance, in an absolute convergent series, you can put all the summands in an other place, and the series will still converge. Thus the convergence is commutative. However, a conditional convegent series is not commutative. There are some other distinctions, but you'll see them soon I guess...

If you're just interested in convergence of series, then there is no need for a distinction between conditional and absolute. However, if you want to do tricky things with the series, then such a distinction is necessairy!

 

[Bazzinga]

Ohh that's interesting! I guess I'll learn all that soon enough, thanks :)