# Asymptotics of $nT(1) + \frac{n}{\lg5}\sum_{i=1}^{\log_5 n}\frac{1}{i}$

I am trying to find asymptotics/running time of recurrence $T(n) = 5T(\frac{n}{5}) + \frac{n}{\lg n}$.
Since Master Theorem for solving the reassurances can't be used, I was able to unroll it and came up with $n(\frac{1}{\ lgn} + \frac{1}{\lg (n/5^2)}+...+\frac{1}{ \ lg5} + 5T(1)) = nT(1) + \frac{n}{\lg5}\sum_{i=1}^{\log_5 n}\frac{1}{i}$

I am not sure how to express the the asymptotics/running time of the last expression.
Thanks!

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Not sure I agree with your unrolling... Anyway, if you browse the site, you will discover that a rigorous method is explained on several pages, which is as follows. Consider $S(k)=T(5^k)/5^k$, then $S(k)=S(k-1)+1/(k\log5)$ hence $S(k)=S(0)+H_k/\log5$, where $H_k=\sum\limits_{i=1}^k1/i$ is the $i$th harmonic number. Note that $T(5^k)=5^kH_k/\log5+5^kT(0)$ and that $H_k/\log k\to1$. Thus, $T(5^k)=\Theta(5^k\log k)$.

Finally, assuming for example that $(T(n))_n$ is monotone, one gets $T(n)=\Theta(n\log\log n)$.

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You would mind pointing me to the example, I can't seems to find it. Thanks – newprint Sep 11 '12 at 22:41
This answer is self contained. – Did Sep 12 '12 at 5:15
shouldn't it be natural logarithm? – Alex Nov 12 '12 at 1:01
First, it is. Second, it does not matter. – Did Nov 12 '12 at 7:27

This is probably just a variation/simplification of Didier's answer. If $T(1)=c_1$ is a constant, then all you need is Harmonic series: $\sum_{k=1}^{n}\frac{1}{k}=O( \ln n)$. Hence, $$S_n = c_1 n + O(n \ln \log_{5}n)=O(n \ln \log_{5}n)$$ since $\lim_{n \to \infty}\frac{n \ln \log_{5}n}{n}=\infty$

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