Monday, 4 February 2019

calculus - Proof of divergence of harmonic series



This question is more about the notation than the actual proof. My professor gave us the following proof:



nk=21knk=2k+1k1xdx=n+121xdx=ln(n+1)ln(2)



I struggle to understand what he did in the first 3 steps. I just don't see how the sum on the left is greater than the sum of the integrals, or why the limits of integration can simply be changed from k to 2 and from k+1 to n+1.



I'd appreciate any help!


Answer




The function f(x)=1x is continuous and decreasing on R+, hence for any kN+ we have
1k>k+1k1xdx


and by summing both sides of (1) for k=1,2,3,,n we get:
Hn=nk=11k>n+11dxx=log(n+1).

In the opposite direction, we may prove a slightly tighter inequality by exploiting 1k<log(2k+12k1) and deducing Hn<log(2n+1).


No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find limh0sin(ha)h without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...