7 days to 2016 #125 Follower Problem

Calculus Level 5

A=n=01n+1k=0n(nk)(1)kk+1B=n=2ζ(n)1nIm([1+i]n[1+in])\Large{A=\sum _{ n=0 }^{ \infty }{ \frac { 1 }{ n+1 } } \sum _{ k=0 }^{ n }{ \left( \begin{matrix} n \\ k \end{matrix} \right) } \frac { { \left( -1 \right) }^{ k } }{ k+1 } \\ B=\sum _{ n=2 }^{ \infty }{ \frac { \zeta \left( n \right) -1 }{ n } { Im }\left( { \left[ 1+i \right] }^{ n }-\left[ 1+{ i }^{ n } \right] \right) } }

If AA can be expressed as

πab\Large{\frac{\pi^{a}}{b}}

And BB can be expressed as

πcd\Large{\frac{\pi^{c}}{d} } .

Find (a+b)(c+d)×12 (a+b)(c+d)\times \frac{1}{2}

Details and Assumption:

Im\text{Im} represents the Imaginary part of a complex number

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