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The reciprocal of the value of lim(nto...

The reciprocal of the value of
`lim_(ntooo) (1-(1)/(2^(2)))(1-(1)/(3^(2)))(1-(1)/(4^(2)))...(1-(1)/(n^(2)))`is__________.

Text Solution

Verified by Experts

The correct Answer is:
`(2)`
We have
`L=underset(ntooo)limprod_(n=2)^(n)(n^(2)-1)/(n^(2))`
`=underset(ntooo)limprod_(n=2)^(n)(n-1)/(n).prod_(n=2)^(n)(n+1)/(n)`
`=underset(ntooo)lim((1)/(2).(2)/(3).(3)/(4)...(n-1)/(n))((3)/(2).(4)/(3).(5)/(4)...(n+1)/(n))`
`=underset(ntooo)lim(1)/(n).(n+1)/(2)=(1)/(2)`
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CENGAGE ENGLISH-LIMITS-Numerical Value Type
  1. The reciprocal of the value of lim(ntooo) (1-(1)/(2^(2)))(1-(1)/(3^(...

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  2. lim(xtooo) f(x)," where "(2x-3)/(x)ltf(x)lt(2x^(2)+5x)/(x^(2))AAxgt0,"...

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  3. If f(x)={x-1,xgeq 1 2x^2-2,x<1,g(x)={x+1,x >0-x^2+1,xlt=0,a n dh(x) ...

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  4. If lim(xtooo) f(x) exists and is finite and nonzero and if lim(xtooo) ...

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  5. If L-("lim")(xvec2)((10-x)^(1/3)-2)/(x-2),t h e nt h ev a l u eof|1(4L...

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  6. If lim(xto0) (p sin2x+(1-cos2x))/(x+tanx)=1, then the value of p is.

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

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

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

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

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

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  12. The value of lim(x to oo ) (log(e)(log(e)x))/(e^(sqrt(x))) is . (a) π...

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  13. about to only mathematics

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

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

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

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

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  19. If lim(xto1) (1+ax+bx^(2))^((e)/((x-1)))=e^(3), then the value of bc i...

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  20. Let f''(x) be continuous at x=0 If lim(xto0) (2f(x)-3af(2x)+bf(8x))/...

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