Limit of an infinite Summation!

Calculus Level 4

\[\Large{ L = \lim_{a \to 0^+} \left[ \sum_{n=0}^\infty \dfrac{(2n+1)a^2 + (-1)^n}{(2n+1)e^{(2n+1)a}} \right]}\]

If \(L\) can be expressed as \(\dfrac{A}{B} \pi^C \) for positive integers \(A,B,C\) such that \(\gcd(A,B) = 1\), then find the value of \(A+B+C\).

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