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Question Number 97984 by abdomathmax last updated on 10/Jun/20

$$\mathrm{calculate}\sum _{\mathrm{k}=0}^{\mathrm{n}}\frac{{(-1)}^{\mathrm{k}}}{2\mathrm{k}+1}{\mathrm{C}}_{\mathrm{n}}^{\mathrm{k}}$$

Answered by mr W last updated on 11/Jun/20

$${(1-x^2)}^n=\underset{k=0}{\sum ^n}{C}_k^n{(-x^2)}^k$$$${(1-x^2)}^n=\underset{k=0}{\sum ^n}{C}_k^n{(-1)}^k{x}^{2k}$$$${\int }_0^1{(1-x^2)}^{n}dx=\underset{k=0}{\sum ^n}{C}_k^n{(-1)}^k{\int }_0^1{x}^{2k}dx$$$${\int }_0^1{(1-x^2)}^{n}dx=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$${\int }_0^{\frac{\pi }{2}}{(1-{\sin }^{2}t)}^n\cos tdt=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$${\int }_0^{\frac{\pi }{2}}{\cos }^{2n+1}tdt=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$$\frac{2n}{2n+1}\times \frac{2n-2}{2n-1}\times ...\times \frac{2}{3}\times {\int }_0^{\frac{\pi }{2}}\cos tdt=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$$\frac{2n}{2n+1}\times \frac{2n-2}{2n-1}\times ...\times \frac{2}{3}\times 1=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$$\frac{\left(2n\right)!!}{(2n+1)!!}=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$$\frac{{[\left(2n\right)!!]}^2}{(2n+1)!}=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$$\frac{{[2^{n}n!]}^2}{(2n+1)!}=\underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n$$$$\Rightarrow \underset{k=0}{\sum ^n}\frac{{(-1)}^k}{2k+1}{C}_k^n=\frac{2^{2n}{(n!)}^2}{(2n+1)!}$$

Commented by mathmax by abdo last updated on 10/Jun/20

$$\mathrm{thank}\mathrm{you}\mathrm{sir}\mathrm{mrw}$$

Answered by mathmax by abdo last updated on 11/Jun/20

$$\mathrm{let}\mathrm{p}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\mathrm{n}}\frac{{(-1)}^{\mathrm{k}}}{2\mathrm{k}+1}{\mathrm{C}}_{\mathrm{n}}^{\mathrm{k}}{\mathrm{x}}^{2\mathrm{k}+1}\mathrm{we}\mathrm{have}$$$${\mathrm{p}}^{{}^{\prime }}\left(\mathrm{x}\right)=\sum _{\mathrm{k}=0}^{\mathrm{n}}{(-1)}^{\mathrm{k}}{\mathrm{C}}_{\mathrm{n}}^{\mathrm{k}}{\mathrm{x}}^{2\mathrm{k}}={(-1)}^{\mathrm{n}}\sum _{\mathrm{k}=0}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}^{\mathrm{k}}{\left({\mathrm{x}}^2\right)}^{\mathrm{k}}{(-1)}^{\mathrm{n}-\mathrm{k}}$$$$={(-1)}^{\mathrm{n}}{({\mathrm{x}}^2-1)}^{\mathrm{n}}={(1-{\mathrm{x}}^2)}^{\mathrm{n}}\Rightarrow \mathrm{p}\left(\mathrm{x}\right)={\int }_0^{\mathrm{x}}{(1-{\mathrm{t}}^2)}^{\mathrm{n}}\mathrm{dt}+\mathrm{c}(\mathrm{c}=0)\mathrm{and}{\mathrm{S}}_{\mathrm{n}}=\mathrm{p}\left(1\right)$$$$={\int }_0^1{(1-{\mathrm{t}}^2)}^{\mathrm{n}}\mathrm{dt}{=}_{\mathrm{t}=\sin \theta }{\int }_0^{\frac{\pi }{2}}{\left({\cos }^2\theta \right)}^{\mathrm{n}}\cos \theta \mathrm{d}\theta ={\int }_0^{\frac{\pi }{2}}{\cos }^{2\mathrm{n}+1}\theta \mathrm{d}\theta =$$$$\mathrm{let}{\mathrm{U}}_{\mathrm{n}}={\int }_0^{\frac{\pi }{2}}{\cos }^{\mathrm{n}}\mathrm{t}\mathrm{dt}\Rightarrow {\mathrm{U}}_{\mathrm{n}}={\int }_0^{\frac{\pi }{2}}{\cos }^{\mathrm{n}-2}\mathrm{t}(1-{\sin }^2\mathrm{t})\mathrm{dt}$$$$={\mathrm{U}}_{\mathrm{n}-2}-{\int }_0^{\frac{\pi }{2}}{\sin }^2\mathrm{t}{\cos }^{\mathrm{n}-2}\mathrm{t}\mathrm{dt}\mathrm{but}\mathrm{by}\mathrm{parts}{\mathrm{f}}^{{}^{\prime }}=\mathrm{sint}{\cos }^{\mathrm{n}-2}\mathrm{and}\mathrm{g}=\mathrm{sint}$$$${\int }_0^{\frac{\pi }{2}}\mathrm{sint}\left({\mathrm{sintcos}}^{\mathrm{n}-2}\mathrm{t}\right)\mathrm{dt}={[-\frac{1}{\mathrm{n}-1}{\cos }^{\mathrm{n}-1}\mathrm{t}\mathrm{sint}]}_0^{\frac{\pi }{2}}+\frac{1}{\mathrm{n}-1}{\int }_0^{\frac{\pi }{2}}\mathrm{cost}{\cos }^{\mathrm{n}-1}\mathrm{t}\mathrm{dt}$$$$=\frac{1}{\mathrm{n}-1}{\mathrm{U}}_{\mathrm{n}}\Rightarrow {\mathrm{U}}_{\mathrm{n}}={\mathrm{U}}_{\mathrm{n}-2}-\frac{1}{\mathrm{n}-1}{\mathrm{U}}_{\mathrm{n}}\Rightarrow (1+\frac{1}{\mathrm{n}-1}){\mathrm{U}}_{\mathrm{n}}={\mathrm{U}}_{\mathrm{n}-2}\Rightarrow$$$$\frac{\mathrm{n}}{\mathrm{n}-1}{\mathrm{U}}_{\mathrm{n}}={\mathrm{U}}_{\mathrm{n}-2}\Rightarrow {\mathrm{U}}_{\mathrm{n}}=\frac{\mathrm{n}-1}{\mathrm{n}}{\mathrm{U}}_{\mathrm{n}-2}\Rightarrow {\mathrm{U}}_{2\mathrm{n}+1}=\frac{2\mathrm{n}}{2\mathrm{n}+1}{\mathrm{U}}_{2\mathrm{n}-1}\Rightarrow$$$$\prod _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{U}}_{2\mathrm{k}+1}=\prod _{\mathrm{k}=1}^{\mathrm{n}}\frac{2\mathrm{k}}{(2\mathrm{k}+1)}\prod _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{U}}_{2\mathrm{k}-1}\Rightarrow$$$${\mathrm{U}}_{2\mathrm{n}+1}={\mathrm{U}}_1\times \prod _{\mathrm{k}=1}^{\mathrm{n}}\frac{2\mathrm{k}}{(2\mathrm{k}+1)}({\mathrm{U}}_1=1)\Rightarrow {\mathrm{S}}_{\mathrm{n}}={\int }_0^{\frac{\pi }{2}}{\cos }^{2\mathrm{n}+1}\mathrm{t}\mathrm{dt}=\prod _{\mathrm{k}=1}^{\mathrm{n}}\frac{2\mathrm{k}}{(2\mathrm{k}+1)}$$

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