Hyberbolics

The series expansions for $\sinh(x)$ and $\cosh(x)$ are $\sinh(x) = \sum_{k=1}^{\infty}\dfrac{x^{2k-1}}{(2k-1)!}$ $\cosh(x) = \sum_{k=0}^{\infty}\dfrac{x^{2k}}{(2k)!}$

The given sum can be written as $\sum_{n=1}^{\infty}\dfrac{2(n + 1)}{(2(n-1))!} = \sum_{n=1}^{\infty}\dfrac{2(n - 1) + 4}{(2(n-1))!} = \sum_{n=2}^{\infty}\dfrac{1}{(2(n-1) - 1)!} + \sum_{n=1}^{\infty} \dfrac{4}{(2(n-1))!}$

Adjusting the limits of both sums, they become $\sum_{n=1}^{\infty}\dfrac{1}{(2n - 1)!} + 4\sum_{n=0}^{\infty} \dfrac{1}{(2n)!}$ which on comparison with the expansions, is equivalent to the expression $\sinh(1) + 4\cosh(1)$ giving $A + B + C = \boxed{6}$