Odd - Even?

Calculus Level 3

Suppose that

A=i=0(12i+112i+2). A = \sum_{i=0}^\infty \left( \frac{1}{ 2i+1} - \frac{1} { 2i + 2 } \right).

What is the value of

i=0(14i+1+14i+312i+2)? \sum_{i=0}^\infty \left(\frac{ 1 } { 4i+1} + \frac{1}{ 4i+3} - \frac{1}{2i+2} \right)?

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