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Question Number 124102 by mnjuly1970 last updated on 30/Nov/20
$$\:\:\:\:\:\:\:\:\:\:\:...{advanced}\:\:\:{calculus}... \\ $$$$\:\:{p}\:,\:{q}\:{are}\:{positive}\:{integers}\:{and} \\ $$$${p}\geqslant{q}\:\::\:{let}\::\phi\left({p},{q}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{p}} \left\{\frac{\mathrm{1}}{{x}}\right\}^{{q}} {dx} \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\phi\left({n},{n}\right)\overset{?} {=}\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\zeta\left({k}+\mathrm{1}\right) \\ $$