Finite sum of reciprocals of odd integers in terms of digamma function

Question

I was reading some of my old notebooks and I came across an (according to me) astounding formula: $$\sum_{k=1}^{n}\frac{1}{2k-1}=\frac{\psi(n+\frac{1}{2})+\gamma}{2}+\log 2$$ I almost never gave proofs then. I tried to prove this. This article tells us that $$\psi(n+\frac{1}{2})=-\gamma-2\log 2+\sum_{k=1}^{n}\frac{2}{2k-1}$$ So this is where it came from. But I don't know how to prove the latter formula. Please try to prove it for me. I may give updates that shall help to prove this.
Any help would be appreciated.