400 Followers Problem - Reciprocal Summations 2

Calculus Level 5

\[\large{S = \left( 1 + \dfrac{1}{2} + \dfrac{1}{3} \right) - \left( \dfrac{1}{4} + \dfrac{1}{5} + \dfrac{1}{6} \right) + \left( \dfrac{1}{7} + \dfrac{1}{8} + \dfrac{1}{9} \right) - \left( \dfrac{1}{10} + \dfrac{1}{11} + \dfrac{1}{12} \right) + \ldots }\]

If \(S\) can be expressed as:

\[\large{\dfrac{A\sqrt{B}}{C}\pi^D + \dfrac{\ln(E)}{F} }\]

for positive integers \(A,B,C,D,E,F\) where \(\gcd(A,C)=1\) and \(B,E\) aren't any \(m^{th}\) power of a positive integer with \(m \in \mathbb Z, \ m \geq 2\).

Submit the value of \(A+B+C+D+E+F\) as your answer.


Also try these:
400 Followers Problem - Reciprocal Summations #1
400 Followers Problem - Reciprocal Summations #3
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