# Partially Accepting

Calculus Level 5

$\large\lim_{n\to\infty} \left(\frac{1/1^4}{1^4} + \frac{1/1^4 + 1/2^4}{2^4} + \ldots +\frac{1/1^4+1/2^4+1/3^4+\ldots+1/n^4}{n^4}\right)$

Given the limit above is equal to $$\frac ab \pi^c$$ for positive integers $$a,b,c$$ with $$a$$ and $$b$$ coprime, find the value of $$a+b+c$$.

Details and Assumptions:

• You are given that $$\displaystyle \sum_{k=1}^\infty \frac1{k^{2n}} = (-1)^{n+1} \frac{B_{2n} \cdot (2\pi)^{2n}}{2(2n)!}$$ where $$B_n$$ is the Bernoulli numbers.

• You may use the following values: $$B_0 = 1, B_2 = \frac16, B_4 = -\frac1{30}, B_6 = \frac1{42}, B_8 = -\frac1{30}$$.

Bonus: Generalize this.

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