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Evalute lim(n->oo)[1/((n+1)(n+2))+1/((n+...

Evalute `lim_(n->oo)[1/((n+1)(n+2))+1/((n+2)(n+4))+......+1/(6n^2)]`

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The given limit is
`L=lim_(ntooo)sum_(r=1)^(n)n . 1/((n+r)(n+2r))`
`=lim_(ntooo)1/n sum_(r=1)^(n)1/((1+r//n)(1+2r//n))`
`=int_(0)^(1)(dx)/((1+x)(1+2x))`
`=int_(0)^(1)((-1)/(1+x)+2/(1+2x))dx`
`=[-log(1+x)+log(1+2x)]_(0)^(1)`
`=[(-log2+log3)-(-log1+log1)]`
`=log(3//2)`
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CENGAGE-DEFINITE INTEGRATION -JEE Advanced Previous Year
  1. Evalute lim(n->oo)[1/((n+1)(n+2))+1/((n+2)(n+4))+......+1/(6n^2)]

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  2. Let f be a non-negative function defined on the interval .[0,1].If int...

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  3. The value of int0^1(x^4(1-x)^4)/(1+x^2)\ dx is

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  4. Let f be a real-valued function defined on the inverval (-1,1) such th...

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  5. The valued of int(sqrt(In2))^(sqrt(In3)) (x sinx^(2))/(sinx^(2)+sin(In...

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  6. Let f:[-1,2]vec[0,oo) be a continuous function such that f(x)=f(1-x)fo...

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  7. Let f:[1/2,1]vecR (the set of all real numbers) be a positive, non-con...

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  8. Let f:[0,2]vecR be a function which is continuous on [0,2] and is diff...

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  9. int((pi)/4)^((pi)/2)(2cosecx)^17 dx

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  10. Let f prime(x)=(192x^3)/(2+sin^4 pix) for all x in RR with f(1/2)=0. I...

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  11. Evaluate: int(-pi//2)^(pi//2)(cosx)/(1+e^x)dx

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  12. If In=int(-pi)^(pi) \ (sinnx)/((1+pi^x) \ sinx) \ dx, n=0,1,2,...... t...

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  13. Let f be a real-valued function defined on interval (0,oo),by f(x)=lnx...

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  14. Let S be the area of the region enclosed by y=e^-x^2,y=0,x=0,a n dx=1....

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  15. Find a for which lim(n->oo) (1^a+2^a+3^a+...+n^a)/((n+1)^(a-1)[(na+1)+...

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  16. Let f:[a,b]to[1,oo) be a continuous function and let g:RtoR be defined...

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  17. Let f:(0,oo) in R be given f(x)=overset(x)underset(1//x)int e^-(t+(1...

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  18. The option(s) with the values of aa n dL that satisfy the following eq...

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  19. Let f(x)=7tan^8x+7tan^6x-3tan^4x-3tan^4x-3tan^2x for all x in (-pi/2,...

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  20. Let f(x)=lim(n->oo)((n^n(x+n)(x+n/2)....(x+n/n))/(n!(x^2+n^2)(x^2+n^2/...

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  21. Let f: Rvec(0,1) be a continuous function. Then, which of the followin...

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