Question Number 225303 by fantastic last updated on 20/Oct/25
![Let S_n (x)=Σ_(r=1) ^n ((sin ((2r−3)2^(−(r+1)) x)cos ((10r+1)2^(−(r+1)) x)−sin( (6r−1)2^(−(r+1)) x)cos ((2r+5)2^(−(r+1)) x))/(2^(r−1) (sin (r2^(3−r) −x)sin (2^(2−r) −x)))) then find the value of lim_(m→∞) (((Σ_(n=0) ^m ∫_0 ^1 cos^(−1) (((cos (x))/(2^n (1+2S_n (x)cos (x)))))dx)/([(d/dx)(Σ_(n=0) ^m cos^(−1) (((cos (x))/(2^n (1+2S_n (x)cos (x)))))]_(x=0) )))](https://www.tinkutara.com/question/Q225303.png)
$${Let} \\ $$$${S}_{{n}} \left({x}\right)=\underset{{r}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{\mathrm{sin}\:\left(\left(\mathrm{2}{r}−\mathrm{3}\right)\mathrm{2}^{−\left({r}+\mathrm{1}\right)} {x}\right)\mathrm{cos}\:\left(\left(\mathrm{10}{r}+\mathrm{1}\right)\mathrm{2}^{−\left({r}+\mathrm{1}\right)} {x}\right)−\mathrm{sin}\left(\:\left(\mathrm{6}{r}−\mathrm{1}\right)\mathrm{2}^{−\left({r}+\mathrm{1}\right)} {x}\right)\mathrm{cos}\:\left(\left(\mathrm{2}{r}+\mathrm{5}\right)\mathrm{2}^{−\left({r}+\mathrm{1}\right)} {x}\right)}{\mathrm{2}^{{r}−\mathrm{1}} \left(\mathrm{sin}\:\left({r}\mathrm{2}^{\mathrm{3}−{r}} −{x}\right)\mathrm{sin}\:\left(\mathrm{2}^{\mathrm{2}−{r}} −{x}\right)\right)} \\ $$$${then}\:{find}\:{the}\:{value}\:{of} \\ $$$$\underset{{m}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\underset{{n}=\mathrm{0}} {\overset{{m}} {\sum}}\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\:\left({x}\right)}{\mathrm{2}^{{n}} \left(\mathrm{1}+\mathrm{2}{S}_{{n}} \left({x}\right)\mathrm{cos}\:\left({x}\right)\right)}\right){dx}}{\left[\frac{{d}}{{dx}}\left(\underset{{n}=\mathrm{0}} {\overset{{m}} {\sum}}\mathrm{cos}^{−\mathrm{1}} \left(\frac{\mathrm{cos}\:\left({x}\right)}{\mathrm{2}^{{n}} \left(\mathrm{1}+\mathrm{2}{S}_{{n}} \left({x}\right)\mathrm{cos}\:\left({x}\right)\right)}\right)\right]_{{x}=\mathrm{0}} \right.}\right) \\ $$