Convergence of ln(k)/k^3 Series

[zeion]

Homework Statement

Determine whether the series converges or diverges.

$$\sum \frac{lnk}{k^3}$$

Homework Equations

The Attempt at a Solution

Since lnk always less than 0, so $$\frac{lnk}{k^3} \leq \frac{1}{k^3}$$ and $$\frac{1}{k^3}$$diverges so$$\frac{lnk}{k^3}$$ diverges.

 

[Mark44]

Homework Statement

Determine whether the series converges or diverges.

$$\sum \frac{lnk}{k^3}$$

Homework Equations

The Attempt at a Solution

Since lnk always less than 0, so $$\frac{lnk}{k^3} \leq \frac{1}{k^3}$$ and $$\frac{1}{k^3}$$diverges so$$\frac{lnk}{k^3}$$ diverges.

I count three mistakes here.
1. Why do you think that ln(k) is always < 0? I'm assuming that k = 1, 2, 3, ...
2. The series whose general term is 1/k³ converges.
3. If you want to show that a given series diverges, its terms must be larger than those of a divergent series. If the terms of the given series are smaller than those of a divergent series, you can't conclude anything.

 

[zeion]

Can I argue that $$ln(k) < k$$ then $$\frac{ln(k)}{k^3} < \frac{k}{k^3}$$then it converge because $$\frac{1}{k^2}$$ converges?

 

[Mark44]

I can buy that. However, your should be able to convince yourself that ln(x) < x.

 

[zeion]

Yeah since ln(x) is the inverse of e^x and they reflect on the y =x.

 

[Mark44]

The fact that y = ln(x) and y = e^x are reflections across the line y = x doesn't prove that ln(x) < x. If y = e^x happens to cross this line a few times means that ln(x) will do so, also.