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If-the-zeta-function-of-2-is-2-n-1-1-n-2-2-2-6-the-sum-of-infinite-rational-numbers-why-converges-for-2-6-an-irra




Question Number 9458 by geovane10math last updated on 09/Dec/16
If the zeta function of 2 is                            𝛇(2) = Σ_(n=1) ^∞  (1/n^2 )                          𝛇(2) = (𝛑^2 /6)  the sum of infinite rational numbers,  why converges for (𝛑^2 /6), an irrational  number?
$$\mathrm{If}\:\mathrm{the}\:\mathrm{zeta}\:\mathrm{function}\:\mathrm{of}\:\mathrm{2}\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\underset{\boldsymbol{{n}}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}^{\mathrm{2}} } \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{\zeta}\left(\mathrm{2}\right)\:=\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}} \\ $$$$\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{infinite}\:\boldsymbol{\mathrm{rational}}\:\mathrm{numbers}, \\ $$$$\mathrm{why}\:\mathrm{converges}\:\mathrm{for}\:\frac{\boldsymbol{\pi}^{\mathrm{2}} }{\mathrm{6}},\:\mathrm{an}\:\boldsymbol{\mathrm{irrational}} \\ $$$$\mathrm{number}? \\ $$
Commented by geovane10math last updated on 09/Dec/16
Know if an infinite serie diverges or  converges is easy...  How calculate the value of serie?
$${Know}\:{if}\:{an}\:{infinite}\:{serie}\:{diverges}\:{or} \\ $$$${converges}\:{is}\:{easy}… \\ $$$${How}\:{calculate}\:{the}\:{value}\:{of}\:{serie}? \\ $$

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