Series involving natural logs: 1/(n*ln(n)*((ln(n))^2 +1)^(1/

[petrol.veem]

I'm trying to decide if two series involving a natural log converge or diverge:

Sum(3,infinity) [ 1 / ( n * ln(n) * ((ln(n))^2 + 1)^1/2 )

and

Sum(1,infinity) [ 1 / (1 + ln(n))

On the first one, i feel it has the "appearance" of an integral test because I can't seem to find any similar series to compare it to. However, I'm not sure how to integrate the associated function, because the (ln(x))^2 always produces an ln(x) in the numerator.

The second one seems easy at first glance to me, noting that 1 / (1 + ln(n)) < 1 / ln(n). But then I am a little bit confused because I'm not sure what to compare 1 / ln(n) with.

 

[tkhunny]

Re: Series involving natural logs

#2 Why would compare to 1/ln(n)? What is your suspicion for 1/[1+ln(n)]? If you believe it diverges, one would do well to compare it to known divergent series. If you are sure that 1/ln(n) diverges, it doesn't do much good to demonstrate the relationship you have shown. Can you show n > 1+ln(n) for n > 1? Now, what do you know about the Harmonic Series?

#1 I may be a little off the wall on this one, but let's see how close we get.

\(\displaystyle \frac{1}{n*ln(n)*\sqrt{[ln(n)]^{2}+1}}\;<\;\frac{1}{n*ln(n)*\sqrt{[ln(n)]^{2}-2ln(n)+1}}\) for n > e

\(\displaystyle \frac{1}{n*ln(n)*\sqrt{[ln(n)]^{2}+1}}\;<\;\frac{1}{n*ln(n)*\sqrt{[ln(n)]^{2}-2ln(n)+1}}\;=\;\frac{1}{n*ln(n)*[ln(n)-1]}\;<\;\frac{1}{n*\sqrt{n}}\) for n > 3322

Are we getting anywhere?

 

[pka]

Re: Series involving natural logs

Also we might consider:
\(\displaystyle \frac{1}{{n\left[ {\ln (n)} \right]\left[ {\sqrt {\left( {\ln (n)} \right)^2 + 1} } \right]}} \leqslant \frac{1}{{n\left[ {\ln (n)} \right]\left[ {\sqrt {\left( {\ln (n)} \right)^2 } } \right]}} \leqslant \frac{1}{{n\left[ {\ln (n)} \right]^2 }}\).

 

[petrol.veem]

Re: Series involving natural logs

Out of curiosity, what would happen if we changed the plus sign to a minus inside the root. Would the series still converge?

\(\displaystyle \frac{1}{{n\left[ {\ln (n)} \right]\left[ {\sqrt {\left( {\ln (n)} \right)^2 - 1} } \right]}}\).

Then the right most series of pka's post would actually be less than the above. But since at infinity the values are so high can the +/-1 be considered "negligible"?

 

[tkhunny]

Re: Series involving natural logs

Out of curiosity, what would happen if we changed the plus sign to a minus inside the root. Would the series still converge?

It is very unlikely that a small change in a constant will make any difference.

since at infinity

I dare you to define that. This suggests to me that you aren't quite getting the idea of a limit. Think on it a little longer.