Is this an indeterminate form?

[Hyacinth42]

I have to determine whether this series converges absolutely, conditionally, or diverges...
$$\sum^{1}_{\infty}\frac{(-1)^{n}}{n*ln(n)}$$

I know it converges conditionally (I have the solution in front of me), but it is kind of vague in one area, it says that lim$$lim_{n\rightarrow\infty}\frac{1}{n*ln(n)}$$ is zero... When you "plug in" infinity for n, you get: $$\frac{1}{\infty*\infty}$$... I know $$\infty*\infty$$ is an indeterminate form, isn't this just a variation of it?

 

[G01]

I have to determine whether this series converges absolutely, conditionally, or diverges...
$$\sum^{1}_{\infty}\frac{(-1)^{n}}{n*ln(n)}$$

I know it converges conditionally (I have the solution in front of me), but it is kind of vague in one area, it says that lim$$lim_{n\rightarrow\infty}\frac{1}{n*ln(n)}$$ is zero... When you "plug in" infinity for n, you get: $$\frac{1}{\infty*\infty}$$... I know $$\infty*\infty$$ is an indeterminate form, isn't this just a variation of it?

An indeterminate form is some expression in which we do not know whether it is converging or diverging, such as:

$$\frac{\infty}{\infty}$$

(The numerator makes the expression diverge, but the denominator is tending to converge to 0. So, we don't know which it actually is doing.)

Another indeterminate form is:

$$0*\infty$$

(Again, the 0 term converges, but the other diverges. So, we cannot tell what the entire expression does.

$$\frac{1}{\infty*\infty}$$ is not an indeterminate form. You can think of that whole quantity in the denominator going to infinity if every term in the denominator is going to infinity. So, your expression basically reduces to:

$$\frac{1}{\infty}$$

which is not an indeterminate form.