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Question Number 33698 by math khazana by abdo last updated on 22/Apr/18
$${let}\:\:{f}_{{n}} \left({x}\right)=\:{n}^{{x}} \:{e}^{−{nx}} \:\:\:{with}\:{x}>\mathrm{0} \\ $$ $$\left.\mathrm{1}\right)\:{study}\:{the}\:{simple}\:{and}\:{uniform}\:{convervence}\:{for} \\ $$ $$\Sigma\:\:{f}_{{n}} \left({x}\right) \\ $$ $$\left.\mathrm{2}\right)\:{let}\:\:{S}\left({x}\right)=\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{f}_{{n}} \left({x}\right).{prove}\:{that} \\ $$ $${S}\left({x}\right)\:\sim\:\frac{\mathrm{1}}{{x}}\:\left(\:{x}\rightarrow\mathrm{0}^{+} \right) \\ $$