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

$$....nicemathematics....$$$$$$$$showthat::$$$$\mathrm{\Phi }={\int }_0^1{li}_2\left(x\right)dx=\frac{{\pi }^2}{6}-1✓$$$$m.n.july.1970$$$$$$

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

$${li}_2\left(x\right)=\mathrm{\Sigma }\frac{x^n}{n^2}$$$${\int }_0^1{li}_2\left(x\right)=\sum _{n⩾1}{\int }_0^1\frac{x^n}{n^2}dx=\sum _{n⩾1}\frac{1}{(n+1)n^2}$$$$=\mathrm{\Sigma }(\frac{1}{n^2}+\frac{1}{n+1}-\frac{1}{n})$$$$=\sum _{n⩾1}\frac{1}{n^2}+\sum _{n⩾1}(\frac{1}{n+1}-\frac{1}{n})=\frac{{\pi }^2}{6}-1$$

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