Does it converge?

Calculus Level 4

$\sum _{x=1 }^{ \infty }{ \frac { { 2 }^{ \frac { x }{ 2 } -\frac { 1 }{ 4 } \cos { \left( \pi x \right) } +\frac { 1 }{ 4 } }{ \pi }^{ \frac { 1 }{ 4 } \cos { \left( \pi x \right) } -\frac { 1 }{ 4 } }\Gamma \left( \frac { x }{ 2 } +1 \right) }{ \Gamma \left( x+1 \right) } }$ can be expressed as $\sqrt { A } +\sqrt { \frac { AB }{ C } } erf\left( \frac { 1 }{ \sqrt { D} } \right)$

Here, A and B are transcendental numbers, C and D are positive integers.

Find the value of $\frac { { B }^{ 4 }-{ A }^{ 4 } }{ 7C }$

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