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Question Number 124102 by mnjuly1970 last updated on 30/Nov/20
           ...advanced   calculus...    p , q are positive integers and  p≥q  : let :φ(p,q)=∫_0 ^( 1) x^p {(1/x)}^q dx    prove that ::                  φ(n,n)=^? 1−(1/(n+1)) Σ_(k=1) ^n ζ(k+1)
$$\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:\:{calculus}… \\ $$$$\:\:{p}\:,\:{q}\:{are}\:{positive}\:{integers}\:{and} \\ $$$${p}\geqslant{q}\:\::\:{let}\::\phi\left({p},{q}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{p}} \left\{\frac{\mathrm{1}}{{x}}\right\}^{{q}} {dx} \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\phi\left({n},{n}\right)\overset{?} {=}\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\zeta\left({k}+\mathrm{1}\right) \\ $$

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