Summing Up The Nearest!

Calculus Level 5

\[\large \displaystyle \sum_{n=1}^{\infty} \frac{1}{\lfloor \sqrt[4]{n} \;\rceil ^7} \]

The above sum can be expressed as \(\frac{a}{b}\pi^{\alpha} + \frac{c}{d}\pi^{\beta} \), where \(a,b,c,d,\alpha ,\) and \(\beta\) are positive integers, with \(a \) and \(b\) being co-prime, \(c\) and \(d\) are co-prime.

Find \( a+b+c+d+\alpha +\beta \).

Details and Assumptions:

\(\bullet \lfloor x \rceil \) denotes the nearest integer to \(x\) .

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