# Highly Nested Summation!

Calculus Level 4

$\large{S = \sum_{i=1}^\infty \sum_{j=1}^\infty \sum_{k=1}^\infty \dfrac{1}{ijk(i+j+k)}}$

If $$S$$ can be expressed as $$\large \dfrac{A}{B} \pi^C$$ for positive integers $$A,B,C$$ with $$\gcd(A,B)=1$$, submit the value of $$A+B+C$$ as your answer.

Bonus: Generalize the expression below in terms of $$n$$:

$\large{P(n) = \sum_{i_1=1}^\infty \sum_{i_2=1}^\infty \cdots \sum_{i_n=1}^\infty \dfrac{1}{i_1 i_2 \dotsm i_n(i_1+i_2+\ldots+i_n)}=\ ?}$

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