Proving Equality in a Summation: A Scientific Approach

[EngWiPy]

Hello,

How to prove this equality:

$$N\,\overline{\gamma}\sum_{k=0}^{N-1}(-1)^k\frac{{N-1\choose k}}{(k+1)^2}=\,\overline{\gamma}\sum_{k=1}^N\frac{1}{k}$$?

Thanks in advance

 

[nicksauce]

It should be relatively easy to prove it by induction.

 

[EngWiPy]

It should be relatively easy to prove it by induction.

The result in the left hand side is my evaluation to an integral using the table of integrals and binomial expansion, and the result in the right hand side obtained from the authors, but I don't know how they did obtain it. The two results are equal and it is easy to proof that by substituting several values of N, but I am wondering if we can go from the LHS equation to the RHS systematically.

Regards