This question is more about the notation than the actual proof. My professor gave us the following proof:
n∑k=21k≥n∑k=2∫k+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 k∈N+ we have
1k>∫k+1k1xdx
and by summing both sides of (1) for k=1,2,3,…,n we get:
Hn=n∑k=11k>∫n+11dxx=log(n+1).
In the opposite direction, we may prove a slightly tighter inequality by exploiting 1k<log(2k+12k−1) and deducing Hn<log(2n+1).
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