Question Number 97984 by abdomathmax last updated on 10/Jun/20
$$\mathrm{calculate}\:\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{2k}+\mathrm{1}}\:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \\ $$
Answered by mr W last updated on 11/Jun/20
![(1−x^2 )^n =Σ_(k=0) ^n C_k ^n (−x^2 )^k (1−x^2 )^n =Σ_(k=0) ^n C_k ^n (−1)^k x^(2k) ∫_0 ^1 (1−x^2 )^n dx=Σ_(k=0) ^n C_k ^n (−1)^k ∫_0 ^1 x^(2k) dx ∫_0 ^1 (1−x^2 )^n dx=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n ∫_0 ^(π/2) (1−sin^2 t)^n cos t dt=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n ∫_0 ^(π/2) cos^(2n+1) t dt=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n ((2n)/(2n+1))×((2n−2)/(2n−1))×...×(2/3)×∫_0 ^(π/2) cos t dt=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n ((2n)/(2n+1))×((2n−2)/(2n−1))×...×(2/3)×1=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n (((2n)!!)/((2n+1)!!))=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n (([(2n)!!]^2 )/((2n+1)!))=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n (([2^n n!]^2 )/((2n+1)!))=Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n ⇒Σ_(k=0) ^n (((−1)^k )/(2k+1))C_k ^n =((2^(2n) (n!)^2 )/((2n+1)!))](https://www.tinkutara.com/question/Q97992.png)
$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{k}} ^{{n}} \left(−{x}^{\mathrm{2}} \right)^{{k}} \\ $$$$\left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} =\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{k}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} {x}^{\mathrm{2}{k}} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{C}_{{k}} ^{{n}} \left(−\mathrm{1}\right)^{{k}} \int_{\mathrm{0}} ^{\mathrm{1}} {x}^{\mathrm{2}{k}} {dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{x}^{\mathrm{2}} \right)^{{n}} {dx}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \:{t}\right)^{{n}} \mathrm{cos}\:{t}\:{dt}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{cos}^{\mathrm{2}{n}+\mathrm{1}} {t}\:{dt}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\frac{\mathrm{2}{n}}{\mathrm{2}{n}+\mathrm{1}}×\frac{\mathrm{2}{n}−\mathrm{2}}{\mathrm{2}{n}−\mathrm{1}}×…×\frac{\mathrm{2}}{\mathrm{3}}×\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{cos}\:{t}\:{dt}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\frac{\mathrm{2}{n}}{\mathrm{2}{n}+\mathrm{1}}×\frac{\mathrm{2}{n}−\mathrm{2}}{\mathrm{2}{n}−\mathrm{1}}×…×\frac{\mathrm{2}}{\mathrm{3}}×\mathrm{1}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\frac{\left(\mathrm{2}{n}\right)!!}{\left(\mathrm{2}{n}+\mathrm{1}\right)!!}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\frac{\left[\left(\mathrm{2}{n}\right)!!\right]^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\frac{\left[\mathrm{2}^{{n}} {n}!\right]^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} \\ $$$$\Rightarrow\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{k}} ^{{n}} =\frac{\mathrm{2}^{\mathrm{2}{n}} \left({n}!\right)^{\mathrm{2}} }{\left(\mathrm{2}{n}+\mathrm{1}\right)!} \\ $$
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
![let p(x) =Σ_(k=0) ^n (((−1)^k )/(2k+1)) C_n ^k x^(2k+1 ) we have p^′ (x) =Σ_(k=0) ^n (−1)^k C_n ^k x^(2k) =(−1)^n Σ_(k=0) ^n C_n ^k (x^2 )^k (−1)^(n−k) =(−1)^n (x^2 −1)^n =(1−x^2 )^n ⇒p(x) =∫_0 ^x (1−t^2 )^n dt+c(c=0) and S_n =p(1) =∫_0 ^1 (1−t^2 )^n dt =_(t =sinθ) ∫_0 ^(π/2) (cos^2 θ)^n cosθ dθ =∫_0 ^(π/2) cos^(2n+1) θ dθ = let U_n =∫_0 ^(π/2) cos^n t dt ⇒U_n =∫_0 ^(π/2) cos^(n−2) t (1−sin^2 t)dt =U_(n−2) −∫_0 ^(π/2) sin^2 t cos^(n−2) t dt but by parts f^′ =sint cos^(n−2) and g =sint ∫_0 ^(π/2) sint(sintcos^(n−2) t)dt =[−(1/(n−1))cos^(n−1) t sint]_0 ^(π/2) +(1/(n−1))∫_0 ^(π/2) cost cos^(n−1) t dt =(1/(n−1))U_n ⇒U_n =U_(n−2) −(1/(n−1))U_(n ) ⇒(1+(1/(n−1)))U_n =U_(n−2) ⇒ (n/(n−1))U_n =U_(n−2) ⇒U_n =((n−1)/n) U_(n−2) ⇒U_(2n+1) =((2n)/(2n+1)) U_(2n−1) ⇒ Π_(k=1) ^n U_(2k+1) =Π_(k=1) ^n ((2k)/((2k+1))) Π_(k=1) ^n U_(2k−1) ⇒ U_(2n+1) =U_1 × Π_(k=1) ^n ((2k)/((2k+1))) (U_1 =1) ⇒S_n =∫_0 ^(π/2) cos^(2n+1) t dt =Π_(k=1) ^n ((2k)/((2k+1)))](https://www.tinkutara.com/question/Q98002.png)
$$\mathrm{let}\:\mathrm{p}\left(\mathrm{x}\right)\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{2k}+\mathrm{1}}\:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{2k}+\mathrm{1}\:} \:\mathrm{we}\:\mathrm{have}\: \\ $$$$\mathrm{p}^{'} \left(\mathrm{x}\right)\:=\sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\left(−\mathrm{1}\right)^{\mathrm{k}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\mathrm{x}^{\mathrm{2k}} \:=\left(−\mathrm{1}\right)^{\mathrm{n}} \sum_{\mathrm{k}=\mathrm{0}} ^{\mathrm{n}} \:\mathrm{C}_{\mathrm{n}} ^{\mathrm{k}} \:\left(\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{k}} \left(−\mathrm{1}\right)^{\mathrm{n}−\mathrm{k}} \\ $$$$=\left(−\mathrm{1}\right)^{\mathrm{n}} \left(\mathrm{x}^{\mathrm{2}} −\mathrm{1}\right)^{\mathrm{n}} \:=\left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} \:\Rightarrow\mathrm{p}\left(\mathrm{x}\right)\:=\int_{\mathrm{0}} ^{\mathrm{x}} \:\left(\mathrm{1}−\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{n}} \:\mathrm{dt}+\mathrm{c}\left(\mathrm{c}=\mathrm{0}\right)\:\mathrm{and}\:\mathrm{S}_{\mathrm{n}} =\mathrm{p}\left(\mathrm{1}\right) \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−\mathrm{t}^{\mathrm{2}} \right)^{\mathrm{n}} \:\mathrm{dt}\:=_{\mathrm{t}\:=\mathrm{sin}\theta} \:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left(\mathrm{cos}^{\mathrm{2}} \theta\right)^{\mathrm{n}} \:\mathrm{cos}\theta\:\mathrm{d}\theta\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{cos}^{\mathrm{2n}+\mathrm{1}} \theta\:\mathrm{d}\theta\:= \\ $$$$\mathrm{let}\:\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{cos}^{\mathrm{n}} \mathrm{t}\:\mathrm{dt}\:\Rightarrow\mathrm{U}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{cos}^{\mathrm{n}−\mathrm{2}} \mathrm{t}\:\left(\mathrm{1}−\mathrm{sin}^{\mathrm{2}} \mathrm{t}\right)\mathrm{dt} \\ $$$$=\mathrm{U}_{\mathrm{n}−\mathrm{2}} −\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{sin}^{\mathrm{2}} \mathrm{t}\:\mathrm{cos}^{\mathrm{n}−\mathrm{2}} \mathrm{t}\:\mathrm{dt}\:\:\mathrm{but}\:\:\:\mathrm{by}\:\mathrm{parts}\:\mathrm{f}^{'} \:=\mathrm{sint}\:\mathrm{cos}^{\mathrm{n}−\mathrm{2}} \:\mathrm{and}\:\mathrm{g}\:=\mathrm{sint} \\ $$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{sint}\left(\mathrm{sintcos}^{\mathrm{n}−\mathrm{2}} \mathrm{t}\right)\mathrm{dt}\:=\left[−\frac{\mathrm{1}}{\mathrm{n}−\mathrm{1}}\mathrm{cos}^{\mathrm{n}−\mathrm{1}} \mathrm{t}\:\mathrm{sint}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:+\frac{\mathrm{1}}{\mathrm{n}−\mathrm{1}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{cost}\:\mathrm{cos}^{\mathrm{n}−\mathrm{1}} \:\mathrm{t}\:\mathrm{dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{n}−\mathrm{1}}\mathrm{U}_{\mathrm{n}} \:\Rightarrow\mathrm{U}_{\mathrm{n}} =\mathrm{U}_{\mathrm{n}−\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{n}−\mathrm{1}}\mathrm{U}_{\mathrm{n}\:} \:\Rightarrow\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}−\mathrm{1}}\right)\mathrm{U}_{\mathrm{n}} =\mathrm{U}_{\mathrm{n}−\mathrm{2}} \:\Rightarrow \\ $$$$\frac{\mathrm{n}}{\mathrm{n}−\mathrm{1}}\mathrm{U}_{\mathrm{n}} =\mathrm{U}_{\mathrm{n}−\mathrm{2}} \:\Rightarrow\mathrm{U}_{\mathrm{n}} =\frac{\mathrm{n}−\mathrm{1}}{\mathrm{n}}\:\mathrm{U}_{\mathrm{n}−\mathrm{2}} \:\Rightarrow\mathrm{U}_{\mathrm{2n}+\mathrm{1}} =\frac{\mathrm{2n}}{\mathrm{2n}+\mathrm{1}}\:\mathrm{U}_{\mathrm{2n}−\mathrm{1}} \:\Rightarrow \\ $$$$\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{U}_{\mathrm{2k}+\mathrm{1}} =\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{2k}}{\left(\mathrm{2k}+\mathrm{1}\right)}\:\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{U}_{\mathrm{2k}−\mathrm{1}} \:\Rightarrow \\ $$$$\mathrm{U}_{\mathrm{2n}+\mathrm{1}} =\mathrm{U}_{\mathrm{1}} ×\:\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{2k}}{\left(\mathrm{2k}+\mathrm{1}\right)}\:\:\left(\mathrm{U}_{\mathrm{1}} =\mathrm{1}\right)\:\Rightarrow\mathrm{S}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\mathrm{cos}^{\mathrm{2n}+\mathrm{1}} \mathrm{t}\:\mathrm{dt}\:=\prod_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{2k}}{\left(\mathrm{2k}+\mathrm{1}\right)} \\ $$