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Question Number 114975 by mnjuly1970 last updated on 22/Sep/20

.... nice mathematics.... show that :: Φ = ∫_0 ^( 1) li_2 (x)dx = (π^2 /6) −1 ✓ m.n.july.1970

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:....\:\:{nice}\:\:{mathematics}....\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:{show}\:\:{that}\:::\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Phi\:=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} {li}_{\mathrm{2}} \left({x}\right){dx}\:=\:\frac{\pi^{\mathrm{2}} }{\mathrm{6}}\:−\mathrm{1}\:\:\checkmark \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970} \\ $$$$ \\ $$

Answered by maths mind last updated on 24/Sep/20

li_2 (x)=Σ(x^n /n^2 ) ∫_0 ^1 li_2 (x)=Σ_(n≥1) ∫_0 ^1 (x^n /n^2 )dx=Σ_(n≥1) (1/((n+1)n^2 )) =Σ((1/n^2 )+(1/(n+1))−(1/n)) =Σ_(n≥1) (1/n^2 )+Σ_(n≥1) ((1/(n+1))−(1/n))=(π^2 /6)−1

$${li}_{\mathrm{2}} \left({x}\right)=\Sigma\frac{{x}^{{n}} }{{n}^{\mathrm{2}} } \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {li}_{\mathrm{2}} \left({x}\right)=\underset{{n}\geqslant\mathrm{1}} {\sum}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}^{{n}} }{{n}^{\mathrm{2}} }{dx}=\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right){n}^{\mathrm{2}} } \\ $$$$=\Sigma\left(\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\frac{\mathrm{1}}{{n}+\mathrm{1}}−\frac{\mathrm{1}}{{n}}\right) \\ $$$$=\underset{{n}\geqslant\mathrm{1}} {\sum}\frac{\mathrm{1}}{{n}^{\mathrm{2}} }+\underset{{n}\geqslant\mathrm{1}} {\sum}\left(\frac{\mathrm{1}}{{n}+\mathrm{1}}−\frac{\mathrm{1}}{{n}}\right)=\frac{\pi^{\mathrm{2}} }{\mathrm{6}}−\mathrm{1} \\ $$

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