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Prove that C(0)-(1)/(3)*C(1)+(1)/(5)*C(2...

Prove that `C_(0)-(1)/(3)*C_(1)+(1)/(5)*C_(2) - …+(-1)^(n)*(1)/(2n+1)C_(n) =(2^(2n)(n !)^(2))/((2n+1)!)`

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AAKASH SERIES-BINOMIAL THEOREM-EXERCISE - 1.4 (Level-2)
  1. Prove that, C(0) *""^(2n)C(n)-C(1)*""^((2n-2))C(n)+C(2) *""^((2n-4))C(...

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  2. If (1+x)^(n)=sum(r=0)^(n)""^(n)C(r )x^(r ) and if sum(r=0)^(n) (1)/(""...

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  3. Prove that C(0)-(1)/(3)*C(1)+(1)/(5)*C(2) - …+(-1)^(n)*(1)/(2n+1)C(n) ...

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  4. If a, b, c, d be 3^(rd), 4^(th), 5^(th) and 6^(th) terms of the expans...

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  5. Show that sum(r=0)^(n) (-1)^(r ). ""^(n)C(r ) {(1)/(2^(r )) +(3^(r ))/...

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  6. If sum(r = 0)^(2n) a(r ) (x-2)^(r ) = sum(r = 0)^(2n) b(r ) (x-3)^(r )...

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  7. Show that the coefficient of x^(n)y^(n) in the expansion of {(1+x)(1+...

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  8. If (1)/(1!(n-1)!) + (1)/(3!(n-3)!) + (1)/(5!(n-5)!) +…. =

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  9. If sum(r =0)^(n) ((r+2)/(r+1)) *""^(n)C(r ) =(255)/(6). Find n.

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  10. Show that 1sum(r =0)^(50) (""^(100)C( r))*(""^(200)C(150+r ))=""^(300)...

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  11. Prove that sum(k=1)^(n-r ) ""^(n-k)C(r )= ""^(n)C( r+1) .

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  12. Show that C(0) +(C(0) +C(1))+(C(0)+C(1)+C(2)) +…+(C(0) +C(1)+…+C(n)) =...

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  13. If x > 0 , write the first negative term in the expansion of (1 + 2x)^...

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  14. If x is small and the expansion of (a)/((1-x)^(2)) +(b)/((2+3x)^(2)) ...

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  15. If x is small and if the expansion of a + (b)/(1 + 2x) + ( c)/(1 -3x^2...

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  16. If (1+2x)/(1-x-x^(2)) =a(0)+a(1)x +a(2)x^(2) +a(3)x^(3)+… then find (a...

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  17. If x is nearly equal to 1 show that (a) (px^(p)-qx^(q))/(p-q)=x^(p+q...

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  18. If |x| lt 1 then 1+n ((2x)/(1+x))+(n(n+1))/(2!) ((2x)/(1=x))^2 + ….......

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  19. Prove the following sum(i=1)^(n) sum(j=1)^(n) sum(k=1)^(n) sum(l=1)...

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  20. Find the value of sum(i=1)^n sum(i=1)^n sum(k=1)^n k

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