Let's do some calculus! (48)

Calculus Level 5

01{1x}2017dx\large \int_0^1 {\left\{ \dfrac 1x \right\}}^{2017} \mathrm{d} x

If the value of the above integral can be represented as

j=1ζ(j+A)B(C+jj)\sum_{j=1}^{\infty} \dfrac{\zeta (j+A) - B}{ \binom{C + j}{j} }

for positive integers A,BA, B and CC, then evaluate A+B+CA+B+C.

Notations:

Hint: Generalize for

01{1x}kdx\int_0^1 {\left\{ \dfrac 1x \right\}}^{k} \mathrm{d} x

where k0k \ge 0 is an integer.


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