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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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