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Question-205367




Question Number 205367 by hardmath last updated on 18/Mar/24
Answered by Ghisom last updated on 18/Mar/24
((n^(3/2) /3^n )+(n^2 /4^n )+(n^(5/2) /5^n ))^(1/n) =  =(n^(3/(2n)) /(60))(20^n +15^n n^(1/2) +12^n n)^(1/n)   n→∞ ⇒  { ((n^(3/(2n)) →1)),(((20^n +...)^(1/n) →20)) :}  ⇒ lim = ((20)/(60))=(1/3)
$$\left(\frac{{n}^{\mathrm{3}/\mathrm{2}} }{\mathrm{3}^{{n}} }+\frac{{n}^{\mathrm{2}} }{\mathrm{4}^{{n}} }+\frac{{n}^{\mathrm{5}/\mathrm{2}} }{\mathrm{5}^{{n}} }\right)^{\mathrm{1}/{n}} = \\ $$$$=\frac{{n}^{\mathrm{3}/\left(\mathrm{2}{n}\right)} }{\mathrm{60}}\left(\mathrm{20}^{{n}} +\mathrm{15}^{{n}} {n}^{\mathrm{1}/\mathrm{2}} +\mathrm{12}^{{n}} {n}\right)^{\mathrm{1}/{n}} \\ $$$${n}\rightarrow\infty\:\Rightarrow\:\begin{cases}{{n}^{\mathrm{3}/\left(\mathrm{2}{n}\right)} \rightarrow\mathrm{1}}\\{\left(\mathrm{20}^{{n}} +…\right)^{\mathrm{1}/{n}} \rightarrow\mathrm{20}}\end{cases} \\ $$$$\Rightarrow\:\mathrm{lim}\:=\:\frac{\mathrm{20}}{\mathrm{60}}=\frac{\mathrm{1}}{\mathrm{3}} \\ $$
Commented by hardmath last updated on 21/Mar/24
thank you dear sir
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{dear}\:\mathrm{sir} \\ $$
Commented by Ghisom last updated on 22/Mar/24
you′re welcome
$$\mathrm{you}'\mathrm{re}\:\mathrm{welcome} \\ $$

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