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the sum of the series ""^(2)C(0) - ""^...

the sum of the series `""^(2)C_(0) - ""^(3)C_(1) x^2 + ""^(4)C_(2) x^(4)-""^(5)C_(3)x^(6) +... ` to ` infty`, is

A

`(1)/(1 + x^(2))^(3)`

B

`(1 - x^(2))^(-3)`

C

`(1)/((1 + x)^(3)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
a
We know that
` (1 - x)^(-n) = ""^(n-1)C_(0) + ""^(n)C_(1) + ""^(n +1)C_(2) x^(2) - ""^(n +1)C_(3) x^(3) + ..........` to `infty` .
Replacing n by 3 and x by `x^(2)` on both sides. We find that
Required sum = `(1 + x^(2))^(-3)` .
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OBJECTIVE RD SHARMA-BINOMIAL THEOREM AND ITS APPLCIATIONS -Section I - Solved Mcqs
  1. If (x^(2) + x)/(1-x) = a(1) x + a(2) x^(2) + ... to infty , |x| lt 1,...

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  2. The sum of the series ""^(4)C(0) + ""^(5)C(1) x + ""^(6)C(2) x^(2)...

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  3. the sum of the series ""^(2)C(0) - ""^(3)C(1) x^2 + ""^(4)C(2) x^(4)...

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  4. If Sn=sum(r=0)^n 1/(nCr) and sum(r=0)^n r/(nCr), then tn/Sn=

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  5. If sn=sum(r < s) (1/(nCr)+1/(nCs)) and tn=sum(r < s)(r/(nCr)+s/(nCs)),...

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  6. The coefficient of x^5 in the expansion of (x^2-x-2)^5 is

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  7. If ar is the coefficient of x^r in the expansion of (1+x+x^2)^n, the...

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  8. If n be a positive integer and Pn denotes the product of the binomial ...

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  9. Prove that in the expansion of (1+x)^n(1+y)^n(1+z)^n , the sum of the ...

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  10. If 1/(sqrt(4x+1)){((1+sqrt(4x+1))/2)^n-((1-sqrt(4x+1))/2)^n}=a0+a1x t...

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  11. If f(x)=x^n ,f(1)+(f^1(1))/1+(f^2(1))/(2!)+(f^n(1))/(n !),w h e r ef^r...

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  12. the coefficient of x^(r) in the expansion of (1 - 4x )^(-1//2), is

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  13. The number of distinct terms in the expansion of (x^(3)+ 1 + (1)/(x...

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  14. If (1 + x + x^(2) + x^(3))^(n)= a(0) + a(1)x + a(2)x^(2) + a(3) x^(3) ...

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  15. The value of ""(n)C(1). X(1 - x )^(n-1) + 2 . ""^(n)C(2) x^(2) (1 -...

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  16. sum(r=1)^(n) {sum(r1=0)^(r-1) ""^(n)C(r) ""^(r)C(r(1)) 2^(r1)} is equ...

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  17. The coefficients of x^(13) in the expansion of (1 - x)^(5) (1 + x...

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  18. If (1+x+x^2)^n=a0+a1x+a2x^2++a(2n)x(2n), find the value of a0+a6++ ,n ...

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  19. The sum of the series 1 + (1)/(1!) ((1)/(4)) + (1.3)/(2!) ((1)/(4))...

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  20. The sum of the series ""^(3)C(0)- ""^(4)C(1) . (1)/(2) + ""^(5)C(2)...

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