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
How to solve this exercise?
0
$\begingroup$
algorithms
-
1For one thing, you have $\log_2\frac ni = \log_2n - \log_2i$. – 2017-01-30
-
1An alternative way is to notice that the sum is an approximation for the integral $n\int_0^1 \log_2(1/x){\rm d}x$. – 2017-01-30
-
0@Arthur Then I come to n^n/n! , but I cannot make it <= 2^n – 2017-01-30
-
0The question [Prove that $\left(\frac{n}{2}\right)^n \gt n! \gt \left(\frac{n}{3}\right)^n \qquad n\ge 6 $](http://math.stackexchange.com/questions/1902324/prove-that-left-fracn2-rightn-gt-n-gt-left-fracn3-rightn-qq) might be useful for that purpose – 2017-01-30