Question Number 160825 by qaz last updated on 07/Dec/21
![lim_(n→∞) [∫_0 ^1 (1+sin ((πt)/2))^n dt]^(1/n) =?](Q160825.png)
$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left[\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}+\mathrm{sin}\:\frac{\pi\mathrm{t}}{\mathrm{2}}\right)^{\mathrm{n}} \mathrm{dt}\right]^{\frac{\mathrm{1}}{\mathrm{n}}} =? \\ $$
Answered by mnjuly1970 last updated on 07/Dec/21
![answer : Ω := sup_( [ 0 ,1 ]) ( 1 +sin (((πt)/2))) =_(is compact set) ^([0 , 1]) 2](Q160828.png)
$${answer}\::\:\:\Omega\::=\:{sup}_{\:\left[\:\mathrm{0}\:,\mathrm{1}\:\right]} \:\left(\:\mathrm{1}\:+{sin}\:\left(\frac{\pi{t}}{\mathrm{2}}\right)\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\underset{{is}\:\:{compact}\:{set}} {\overset{\left[\mathrm{0}\:,\:\mathrm{1}\right]} {=}}\:\:\:\:\:\mathrm{2}\:\: \\ $$