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Question Number 67519 by mathmax by abdo last updated on 28/Aug/19

if (z/(e^z −1)) =Σ_(n=0) ^∞  B_n  (z^n /(n!))  1) calculate B_0 ,B_1 ,B_2 ,B_3 ,B_4   2)prove that z→(1/(e^z −1))+(1/2) is a odd function  conclude that  B_(2n+1) =0  for n≥1

$${if}\:\frac{{z}}{{e}^{{z}} −\mathrm{1}}\:=\sum_{{n}=\mathrm{0}} ^{\infty} \:{B}_{{n}} \:\frac{{z}^{{n}} }{{n}!} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{B}_{\mathrm{0}} ,{B}_{\mathrm{1}} ,{B}_{\mathrm{2}} ,{B}_{\mathrm{3}} ,{B}_{\mathrm{4}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:{z}\rightarrow\frac{\mathrm{1}}{{e}^{{z}} −\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{2}}\:{is}\:{a}\:{odd}\:{function}\:\:{conclude}\:{that} \\ $$$${B}_{\mathrm{2}{n}+\mathrm{1}} =\mathrm{0}\:\:{for}\:{n}\geqslant\mathrm{1} \\ $$

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