Why does the Big O notation of summation give log I’m not understanding the following line in a proof
I believe it should be $O(h^2\psi(2h))$ since $\sum_{j=1}^{h}\frac{1}{j}\leqslant h\cdot1=h$

This is but a harmonic series summation:
$$\sum_{j=1}^h\frac{2h\psi(2h)}j=2h\psi(2h)\sum_{j=1}^h\frac1j\in2h\psi(2h)\cdot O(\log h)\sim O(h\psi(2h)\log h)$$
