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Theorem 3.29 If $p >1$ then $\sum_{n=2}^{\infty} \frac{1}{n(\log n)^p}$ converges; if $p \leq 1$, the series diverges.

Proof:

The monotonicity of the logarithmic function implies that $\{\log n\}$ increases. Hence $\frac{1}{n\log n}$ decreases, and we can apply theorem 3.27; this leads us to the series

$\sum_{k=1}^{\infty} 2^k \frac{1}{2^k(\log 2^k)^p}= \sum_{k=1}^{\infty} \frac{1}{(k\log 2)^p}= \frac{1}{(\log 2)^p} \sum_{k=1}^{\infty} \frac{1}{k^p}$ and the conclusion follows from Theorem 3.28.

I understand the proof thus far, but Rudin goes on to say that this procedure may evidently be continued. For instance,

$\sum_{k=3}^{\infty} \frac{1}{n\log n(\log\log n)}$ diverges, whereas

$\sum_{k=3}^{\infty} \frac{1}{n\log n(\log\log n)^2}$ converges.

I want to continue the procedure to show for which $p$ does $\sum_{k=3}^{\infty} \frac{1}{n\log n(\log\log n)^p}$ converge.

Here are the relevant theorems:

Theorem 3.27 Suppose $a_1 \geq a_2 \geq ... \geq 0$. Then the series $\sum_{k=1}^{\infty} a_n$ converges if and only if $\sum_{k=1}^{\infty} 2^n a_{2^n}$ converges.

Theorem 3.28 $\sum \frac{1}{n^p}$ converges if $p>1$ and diverges if $p \leq 1$.

Here is what I have so far:

By Theorem 3.27 $\sum_{k=3}^{\infty} \frac{1}{n\log n(\log\log n)^p}$ converges if and only if $\sum_{k=2}^{\infty}2^n \frac{1}{2^n\log 2^n(\log\log 2^n)^p} = \frac{1}{\log 2} \sum_{k=2}^{\infty} \frac{1}{n(\log\log 2^n)^p}$

Now I'm stuck. How may the procedure be continued? Please note it is my goal to show a result similar to Theorem 3.29. It is not my goal to simply show that $\sum_{k=3}^{\infty} \frac{1}{n\log n(\log\log n)^p}$ converges for some $p$. Any hint would be greatly appreciated. Thank you.

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    $\begingroup$ $\log (\log 2^n) = \log (n\log 2) = \log n + \log (\log 2)$. Ignore the first few terms maybe, then you have $\frac{1}{2}\log n < \log (\log 2^n) < \log n$ (since $\log (\log 2) < 0$). $\endgroup$ Oct 30, 2014 at 22:01
  • $\begingroup$ @DanielFischer This really works! Thank you very much. $\endgroup$
    – Henry Choi
    Aug 10, 2020 at 9:04

2 Answers 2

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I think I figured it out.

By Theorem 3.29 if $p>1$, then $\Sigma_{n=2}^{\infty}\frac{1}{n(logn)^p}$ converges.

By Theorem 3.27 $\Sigma_{n=3}^{\infty} \frac{1}{nlogn(loglogn)^p}$ converges if and only if $\frac{1}{log2} \Sigma_{n=2}^{\infty} \frac{1}{n(loglog2^n)^p}$ converges.

But $\frac{1}{n(loglog2^n)^p} = \frac{1}{n(logn+loglog2)^p} \leq \frac{1}{n(logn)^p}$

Hence $\frac{1}{log2} \Sigma_{n=2}^{\infty} \frac{1}{n(loglog2^n)^p}$ converges for $p>1$, and therefore $\Sigma_{n=3}^{\infty} \frac{1}{nlogn(loglogn)^p}$ converges for $p>1$.

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    $\begingroup$ "$\frac{1}{n(logn+loglog2)^p} \leq \frac{1}{n(logn)^p}$" I don't think this is true, since if you let $n=10$, $p=1.1$, then $\frac{1}{n(logn+loglog2)^p} \approx 0.048 $ and $\frac{1}{n(logn)^p} \approx 0.040 $ $\endgroup$
    – ignoramus
    Nov 1, 2014 at 8:51
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Denoting by $\ln^{[t]}$ the $t$th iteration of logarithm, one can get $$\frac{1}{\ln^{[t]} n}-\frac{1}{\ln^{[t]} (n+1)}=\frac{1}{n(\ln n)\ldots\left(\ln^{[t]}n\right)\left(\ln^{[t+1]}n\right)^2}+O\left(\frac{1}{n^2}\right),$$ so that $$\sum\limits_{n=N}^\infty \frac{1}{n(\ln n)\ldots\left(\ln^{[t]}n\right)\left(\ln^{[t+1]}n\right)^2}\leqslant\sum\limits_{n=N}^\infty\left(\frac{1}{\ln^{[t]} n}-\frac{1}{\ln^{[t]} (n+1)}\right)+c=\frac{1}{\ln^{[t]} N}+c,$$ which proves the first part of the result. The other part is done just by noting that $$\frac{d\,\ln^{[t]}x}{dx}=\frac{1}{x(\ln x)\ldots\left(\ln^{[t-1]}x\right)},$$ because $$\sum\limits_{n=u}^{v}\frac{1}{n(\ln n)\ldots\left(\ln^{[t-1]}n\right)}\geqslant\int\limits_{u+1}^{v}\frac{dx}{x(\ln x)\ldots\left(\ln^{[t-1]}x\right)}.$$

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  • $\begingroup$ This looks very interesting, but I'm having trouble understanding it. Could you explain how the results follow from your two statements? Thanks $\endgroup$
    – ignoramus
    Nov 1, 2014 at 10:24
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    $\begingroup$ @ignoramus I added the details. $\endgroup$
    – user2097
    Nov 1, 2014 at 11:39
  • $\begingroup$ Ah ok I see, thanks! $\endgroup$
    – ignoramus
    Nov 1, 2014 at 12:02

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