Question and Answers Forum

All Questions      Topic List

Vector Questions

Previous in All Question      Next in All Question      

Previous in Vector      Next in Vector      

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

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com