Question Number 227026 by Mohammedqasim last updated on 26/Dec/25
Answered by Kassista last updated on 26/Dec/25
$$ \\ $$$${Numerator}: \\ $$$$\frac{{d}}{{dx}}\left(\pi{x}+\sqrt[{\mathrm{5}}]{\mathrm{99}}\right)=\pi \\ $$$$ \\ $$$${Denominator}: \\ $$$$\int_{−\mathrm{2}} ^{\:\mathrm{2}} \frac{{x}^{\mathrm{3}} +\mathrm{sin}\:{x}}{{x}^{\mathrm{2}} +\mathrm{cos}\:{x}}{dx}\:\overset{{odd}\:{function}} {\rightarrow}\:=\:\mathrm{0} \\ $$$$ \\ $$$$\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}=\:\mathrm{2}\underset{{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\mathrm{1}}{\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}}\right)+\left(\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{3}}\right)+\left(\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{4}}\right)+…+\left(\frac{\mathrm{1}}{{n}}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\right)= \\ $$$$ \\ $$$$=\mathrm{2}\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}=\mathrm{2} \\ $$$$ \\ $$$${Putting}\:{all}\:{together}: \\ $$$$\mathrm{10}{e}^{\frac{{i}\pi}{\mathrm{2}}} =\mathrm{10}\left(\mathrm{cos}\left(\mathrm{90}^{\mathrm{0}} \right)+{i}\:\mathrm{sin}\left(\mathrm{90}^{\mathrm{0}} \right)\right)=\mathrm{10}\left(\mathrm{0}+{i}\centerdot\mathrm{1}\right)=\mathrm{10}{i} \\ $$$$ \\ $$