taylor expansion – Calculating a series (finding a closed form)

I would like to find the value of $$v=sum_{ineq2k-1}frac{(-1)^i}{(2k-1)i}$$.
The thing I can think of is to separate them into two parts namely
$$frac12sum_{k,linmathbb{Z^+}}frac{1}{(2k-1)l}-sum_{kneq l}frac{1}{(2k-1)(2l-1)}$$.
And this reminds me of the Taylor series which appears to be the alternating harmonic series and the alternating series of the reciprocals of the odd numbers. Is there any way?