Question Number 216425 by universe last updated on 07/Feb/25
Commented by universe last updated on 07/Feb/25
$${find}\:{sum} \\ $$
Answered by MrGaster last updated on 08/Feb/25
![=∫_0 ^(π/2) Σ_(n=1) ^(10) ((sin^2 (3^n x))/(cos(3^(n+1) x)))dx =∫_0 ^(π/2) (((sin^2 (3x))/(cos(9x)))+((sin^2 (9x))/(cos(27x)))+…+((sin^2 (3^(10) x))/(cos(3^(11) x))))dx =Σ_(n=1) ^(10) ∫_0 ^(π/2) ((sin^2 (3^n x))/(cos(^(3n+1) x)))dx =Σ_(n=1) ^(10) [(1/3^n )∫_0 ^(3^n (π/2)) ((sin^2 u)/(cos(3u)))du] =Σ_(n=1) ^(10) [(1/3^n )∫_0 ^(3^n (π/2)) ((sin^2 u)/(cos(3u)))du] =Σ_(n=1) ^(10) [(1/3^n )((1/6)ln∣((1+sin(3u))/(1−sin(3u))∣−(1/3)tan(3u)]_0 ^(3^n (π/2)) =Σ_(n=1) ^(10) [(1/3^n )((1/6)ln∣((1+sin(3^(n+1) (π/2)))/(1−sin(3^(n+1) (π/2))))∣−(1/3)tan(3^(n+1) (π/2)))] =Σ_(n=1) ^(10) [(1/3^n )((1/6)ln∣((1+(−1)^(n+1) )/(1−(−1)^(n+1) ))∣−(1/3)tan(3^(n+1) (π/2)))] =Σ_(n=1) ^(10) [(1/3^n )((1/6)ln∣(2/(“0”))∣−(1/3)tan(3^(n+1) (π/2)))] =Σ_(n=1) ^(10) [(1/3^n )((1/6)ln∣“∞”∣−(1/3)tan(3^(n+1) (π/2)))] Next some“ illegal” operations will becarried out: =Σ_(n=1) ^(10) [(1/3^n )((∞/6)−(∞/3))] =Σ_(n=1) ^(10) [(1/3^n )((∞/6)−((2∞)/6))] =Σ_(n=1) ^(10) [(1/3^n )(−(∞/6))] =Σ_(n=1) ^(10) [−(∞/(6∙3^n ))] =−(∞/6)Σ_(n=1) ^(10) (1/3^n ) =(∞/6)∙(1/2)(1−(1/3^(10) )) =−(∞/(12))(1−(1/3^(10) )) determinant (((−(∞/(12))≈^? “0.9113919361034865649028795807632075543645…”)))](https://www.tinkutara.com/question/Q216436.png)
$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\frac{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}^{{n}} {x}\right)}{\mathrm{cos}\left(\mathrm{3}^{{n}+\mathrm{1}} {x}\right)}{dx} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\frac{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}{x}\right)}{\mathrm{cos}\left(\mathrm{9}{x}\right)}+\frac{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{9}{x}\right)}{\mathrm{cos}\left(\mathrm{27}{x}\right)}+\ldots+\frac{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}^{\mathrm{10}} {x}\right)}{\mathrm{cos}\left(\mathrm{3}^{\mathrm{11}} {x}\right)}\right){dx} \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{3}^{{n}} {x}\right)}{\mathrm{cos}\left(^{\mathrm{3}{n}+\mathrm{1}} {x}\right)}{dx} \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\int_{\mathrm{0}} ^{\mathrm{3}^{{n}} \frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}^{\mathrm{2}} {u}}{\mathrm{cos}\left(\mathrm{3}{u}\right)}{du}\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\int_{\mathrm{0}} ^{\mathrm{3}^{{n}} \frac{\pi}{\mathrm{2}}} \frac{\mathrm{sin}^{\mathrm{2}} {u}}{\mathrm{cos}\left(\mathrm{3}{u}\right)}{du}\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(\frac{\mathrm{1}}{\mathrm{6}}\mathrm{ln}\mid\frac{\mathrm{1}+\mathrm{sin}\left(\mathrm{3}{u}\right)}{\mathrm{1}−\mathrm{sin}\left(\mathrm{3}{u}\right.}\mid−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{tan}\left(\mathrm{3}{u}\right)\right]_{\mathrm{0}} ^{\mathrm{3}^{{n}} \frac{\pi}{\mathrm{2}}} \right. \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(\frac{\mathrm{1}}{\mathrm{6}}\mathrm{ln}\mid\frac{\mathrm{1}+\mathrm{sin}\left(\mathrm{3}^{{n}+\mathrm{1}} \frac{\pi}{\mathrm{2}}\right)}{\mathrm{1}−\mathrm{sin}\left(\mathrm{3}^{{n}+\mathrm{1}} \frac{\pi}{\mathrm{2}}\right)}\mid−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{tan}\left(\mathrm{3}^{{n}+\mathrm{1}} \frac{\pi}{\mathrm{2}}\right)\right)\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(\frac{\mathrm{1}}{\mathrm{6}}\mathrm{ln}\mid\frac{\mathrm{1}+\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }{\mathrm{1}−\left(−\mathrm{1}\right)^{{n}+\mathrm{1}} }\mid−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{tan}\left(\mathrm{3}^{{n}+\mathrm{1}} \frac{\pi}{\mathrm{2}}\right)\right)\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(\frac{\mathrm{1}}{\mathrm{6}}\mathrm{ln}\mid\frac{\mathrm{2}}{“\mathrm{0}''}\mid−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{tan}\left(\mathrm{3}^{{n}+\mathrm{1}} \frac{\pi}{\mathrm{2}}\right)\right)\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(\frac{\mathrm{1}}{\mathrm{6}}\mathrm{ln}\mid“\infty''\mid−\frac{\mathrm{1}}{\mathrm{3}}\mathrm{tan}\left(\mathrm{3}^{{n}+\mathrm{1}} \frac{\pi}{\mathrm{2}}\right)\right)\right] \\ $$$$\mathrm{Next}\:\mathrm{some}“\:\mathrm{illegal}''\:\mathrm{operations}\:\mathrm{will}\: \\ $$$$\mathrm{becarried}\:\mathrm{out}: \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(\frac{\infty}{\mathrm{6}}−\frac{\infty}{\mathrm{3}}\right)\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(\frac{\infty}{\mathrm{6}}−\frac{\mathrm{2}\infty}{\mathrm{6}}\right)\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[\frac{\mathrm{1}}{\mathrm{3}^{{n}} }\left(−\frac{\infty}{\mathrm{6}}\right)\right] \\ $$$$=\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\left[−\frac{\infty}{\mathrm{6}\centerdot\mathrm{3}^{{n}} }\right] \\ $$$$=−\frac{\infty}{\mathrm{6}}\underset{{n}=\mathrm{1}} {\overset{\mathrm{10}} {\sum}}\frac{\mathrm{1}}{\mathrm{3}^{{n}} } \\ $$$$=\frac{\infty}{\mathrm{6}}\centerdot\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{10}} }\right) \\ $$$$=−\frac{\infty}{\mathrm{12}}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{10}} }\right) \\ $$$$\begin{array}{|c|}{−\frac{\infty}{\mathrm{12}}\overset{?} {\approx}“\mathrm{0}.\mathrm{9113919361034865649028795807632075543645}\ldots''}\\\hline\end{array} \\ $$
Answered by MathematicalUser2357 last updated on 25/Feb/25
$$\downarrow\downarrow\downarrow \\ $$
Commented by MathematicalUser2357 last updated on 25/Feb/25
sin^2(3 x) sec(9 x) + sin^2(9 x) sec(27 x) + sin^2(27 x) sec(81 x) + sin^2(81 x) sec(243 x) + sin^2(243 x) sec(729 x) + sin^2(729 x) sec(2187 x) + sin^2(2187 x) sec(6561 x) + sin^2(6561 x) sec(19683 x) + sin^2(19683 x) sec(59049 x) + sin^2(59049 x) sec(177147 x)