# Who's up to the challenge? 37

**Calculus**Level 5

\[ \lim _{a\rightarrow1^-}{\large \displaystyle\sum _{ k=1 }^{ \infty }{ \sum _{ n=1 }^{ \infty }{ { (-1) }^{ k+n }\dfrac { { H }_{ k }{ H }_{ n } }{ k+n }a^{k+n} } } }=\dfrac { \zeta (a) }{ b } -\dfrac { (\ln d)^c }{ f } +\dfrac { (\ln h)^g }{ i } +j\ln { k } -l\]

The equation above holds true, where \(H_n\) denotes the \(n^{\text{th}}\) harmonic number and \(a,b,c,\ldots,k,l\) are positive integers.

Find \(a+b+c+d+f+g+h+i+j+k+l-1\).

**Notation**: \(\zeta(\cdot) \) denotes the Riemann zeta function.