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Let S(n)=1+2+3+...+n " and " P(n)=(S(2))...

Let `S_(n)=1+2+3+...+n " and " P_(n)=(S_(2))/(S_(2)-1).(S_(3))/(S_(3)-1).(S_(4))/(S_(4)-1)...(S_(n))/(S_(n)-1)`, where `n inN,(nge2) Then ``underset(ntooo)limP_(n)`=__________.

A

`(1)/(4)`

B

`(1)/(24)`

C

`(1)/(16)`

D

`(1)/(8)`

Text Solution

Verified by Experts

The correct Answer is:
`(3)`
`S_(n)=(n(n+1))/(n)" and "S_(n)-1=((n+2)(n-1))/(2)`
`:." "(S_(n))/(S_(n)-1)=(n(n+1))/(2).(2)/((n+2)(n-1))=((n)/(n-1))((n+1)/(n+2))`
`:." "P_(n)=((2)/(1).(3)/(2).(4)/(3).(5)/(4)...(n)/(n-1))((3)/(4).(4)/(5).(5)/(6)...(n+1)/(n+2))`
`=((n)/(1))((3)/(n+2))`
`:." "underset(ntooo)limP_(n)=3`
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CENGAGE-LIMITS-Exercise (Numerical)
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  3. If lim(xto0) (p sin2x+(1-cos2x))/(x+tanx)=1, then the value of p is.

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  4. The value of lim(xtooo) ((100)/(1-x^(100))-(50)/(1-x^(50))) is .

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  5. If L= lim(xto2) (root(3)(60+x^(2))-4)/(sin(x-2)), then the value of 1/...

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  6. The value of lim(xtooo) ((20^(x)-1)/(19(5^(x))))^(1//x) is .

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  7. The value of lim(ntooo) [root(3)((n+1)^(2))-root(3)((n-1)^(2))] is .

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  8. If L=-lim(ntooo) (2xx3^(2)xx2^(3)xx3^(4)...xx2^(n-1)xx3^(n))^((1)/((n^...

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  10. If lim(xto0) (1-sqrt(cos2x).root(3)(cos3x).root(4)(cos4x)...root(n)(co...

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  11. The value of lim(x to oo ) (x-x^(2)log(e)(1+(1)/(x))) is .

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  12. Let S(n)=1+2+3+...+n " and " P(n)=(S(2))/(S(2)-1).(S(3))/(S(3)-1).(S(4...

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  13. If lim(xto1)(asin(x-1)+bcos(x-1)+4)/(x^(2)-1)=-2, then |a+b| is.

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  14. Let lim(xto1) (x^(a)-ax+a-1)/((x-1)^(2))=f(a). Then the value of f(4) ...

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  15. Number of integral values of k for which lim(xto1) sin^(-1)((k)/(log...

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  19. The integer n for which lim(xto0) ((cosx-1)(cosx-e^(x)))/(x^(n)) is a ...

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