Show that the summation $$\sum_{i=1}^n log_2 \frac{n}{i}$$ is O(n). You may assume that n is a power of 2.
Hint: Use induction to reduce the problem to that for n/2
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1For one thing, you have $\log_2\frac ni = \log_2n - \log_2i$. – Arthur Jan 30 '17 at 19:46
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1An alternative way is to notice that the sum is an approximation for the integral $n\int_0^1 \log_2(1/x){\rm d}x$. – Winther Jan 30 '17 at 19:57
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@Arthur Then I come to n^n/n! , but I cannot make it <= 2^n – MongoliaOba Jan 30 '17 at 20:17
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The question Prove that $\left(\frac{n}{2}\right)^n \gt n! \gt \left(\frac{n}{3}\right)^n \qquad n\ge 6 $ might be useful for that purpose – Winther Jan 30 '17 at 20:25