Question Number 197292 by universe last updated on 12/Sep/23
$$\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\int_{\mathrm{0}\:} ^{\mathrm{1}} \frac{{nx}^{{n}−\mathrm{1}} }{\mathrm{1}+{x}}{dx}\:\:=\:\:\:? \\ $$
Answered by witcher3 last updated on 12/Sep/23
![2x≤1+x≤2,∀x∈[0,1] ⇒((nx^(n−1) )/2)≤((nx^(n−1) )/(1+x))≤((nx^(n−2) )/2) (n/(2n))=(1/2)≤∫_0 ^1 ((nx^(n−1) )/(1+x))dx≤(n/(2(n−1))) ⇒lim_(n→∞) ∫_0 ^1 nx^(n−1) (1+x)^(−1) dx=(1/2)](https://www.tinkutara.com/question/Q197293.png)
$$\mathrm{2x}\leqslant\mathrm{1}+\mathrm{x}\leqslant\mathrm{2},\forall\mathrm{x}\in\left[\mathrm{0},\mathrm{1}\right] \\ $$$$\Rightarrow\frac{\mathrm{nx}^{\mathrm{n}−\mathrm{1}} }{\mathrm{2}}\leqslant\frac{\mathrm{nx}^{\mathrm{n}−\mathrm{1}} }{\mathrm{1}+\mathrm{x}}\leqslant\frac{\mathrm{nx}^{\mathrm{n}−\mathrm{2}} }{\mathrm{2}} \\ $$$$\frac{\mathrm{n}}{\mathrm{2n}}=\frac{\mathrm{1}}{\mathrm{2}}\leqslant\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{nx}^{\mathrm{n}−\mathrm{1}} }{\mathrm{1}+\mathrm{x}}\mathrm{dx}\leqslant\frac{\mathrm{n}}{\mathrm{2}\left(\mathrm{n}−\mathrm{1}\right)} \\ $$$$\Rightarrow\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{nx}^{\mathrm{n}−\mathrm{1}} \left(\mathrm{1}+\mathrm{x}\right)^{−\mathrm{1}} \mathrm{dx}=\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$ \\ $$
Answered by witcher3 last updated on 12/Sep/23
![Meth 2 by part =(x^n /(1+x))]_0 ^1 +∫_0 ^1 (x^n /((1+x)^2 ))dx=(1/2)+∫_0 ^1 (x^n /((1+x)^2 ))dx by part 2 ∫_0 ^1 x^n (1+x)^2 dx=(1/(4(n+1)))+(2/(n+1))∫_0 ^1 (x^(n+1) /((1+x)^3 ))dx→0 integral→(1/2)](https://www.tinkutara.com/question/Q197294.png)
$$\mathrm{Meth}\:\mathrm{2}\: \\ $$$$\mathrm{by}\:\mathrm{part} \\ $$$$\left.=\frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}+\mathrm{x}}\right]_{\mathrm{0}} ^{\mathrm{1}} +\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} }{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx}=\frac{\mathrm{1}}{\mathrm{2}}+\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} }{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} }\mathrm{dx} \\ $$$$\mathrm{by}\:\mathrm{part}\:\mathrm{2} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{x}^{\mathrm{n}} \left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{2}} \mathrm{dx}=\frac{\mathrm{1}}{\mathrm{4}\left(\mathrm{n}+\mathrm{1}\right)}+\frac{\mathrm{2}}{\mathrm{n}+\mathrm{1}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}+\mathrm{1}} }{\left(\mathrm{1}+\mathrm{x}\right)^{\mathrm{3}} }\mathrm{dx}\rightarrow\mathrm{0} \\ $$$$\mathrm{integral}\rightarrow\frac{\mathrm{1}}{\mathrm{2}} \\ $$