Determine the Convergence of E(ln(n)/n3) and Justify Your Answer

[IntegrateMe]

Determine if the series converges absolutely, conditionally, or if it diverges. Give justifications for your answers.

(note: "E" is the sigma notation)

1. E from 1 to infinity of ln(n)/n³

Ok, i basically used the limit comparison test and pulled out a 1/n³. As we know by p-series, 1/n³ converges and now becomes our "bn."

So, using the LCT i get... limit going to infinity of an/bn = lim to infinity of [ln(n)/n³]/(1/n³) which results in the lim to infinity of ln(n) which = infinity. Because our limit is infinity it means that if bn diverges, an diverges as well. But in this case, bn converged! Any thoughts??

Because bn converged i thought the answer would be "converges conditionally" but i was marked points off because the answer is "converges absolutely."

 

[Coto]

Determine if the series converges absolutely, conditionally, or if it diverges. Give justifications for your answers.

(note: "E" is the sigma notation)

1. E from 1 to infinity of ln(n)/n³

Ok, i basically used the limit comparison test and pulled out a 1/n³. As we know by p-series, 1/n³ converges and now becomes our "bn."

So, using the LCT i get... limit going to infinity of an/bn = lim to infinity of [ln(n)/n³]/(1/n³) which results in the lim to infinity of ln(n) which = infinity. Because our limit is infinity it means that if bn diverges, an diverges as well. But in this case, bn converged! Any thoughts??

Because bn converged i thought the answer would be "converges conditionally" but i was marked points off because the answer is "converges absolutely."

I would just show that,

$$\log n < n \implies \frac{\log n}{n^3} < \frac{1}{n^2}.$$

What do you know about the latter series?

 

[IntegrateMe]

Coto, how does that prove the series converges absolutely? And why did you pull a 1/n^2 out instead of a 1/n^3

 

[Coto]

Hey IntegrateMe,

Well first off, what does it mean for a series to absolutely converge?

Second, what do you know about:

$$\sum_1 ^{\infty} \frac{1}{n^2}.$$

Does it converge?

Lastly,
$$\log n < n \implies \frac{\log n}{n^3} < \frac{n}{n^3} = \frac{1}{n^2},$$

and for $$n \geq 1, \log n \geq 0.$$

Hopefully this leads you in the right direction.

 

[IntegrateMe]

Thanks Coto!

I was just trying to gain some intuition; you helped a lot and i understand convergence a lot better now.