Can someone explain how Big-Oh works with Summations?

Question

I know this isn\'t strictly a programming question, but it is a computer science question so I\'m hoping someone can help me.

I\'ve been working on my Algor

Answer 1

http://en.wikipedia.org/wiki/Big_O_notation

N repetitions of g(m)=O(f(m)) is O(N*f(m)) for any f(N).

Sum of i=1..N of i*g(i) is O(N*f(N)) if g(n)=O(f(n)) and f is monotonic.

Definition: g(n)=O(f(n)) if some c,m exist so for all n>m, g(n)<=c*f(n)

The sum is for i=1..N of i*f(i).

If f is monotonic in i this means every term is <= if(N) <= Nf(N). So the sum is less than N*c*f(N) so the sum is O(N*f(N)) (witnessed by the same c,m that makes g(n)=O(f(n)))

Of course, log_2(x) and x^2 are both monotonic.