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lim(n to oo)[(n)/(1+n^(2))+(n)/(4+n^(2))...

`lim_(n to oo)[(n)/(1+n^(2))+(n)/(4+n^(2))+(n)/(9+n^(2))+…+(1)/(2n^2)]=`

A

`(pi)/(2)`

B

`(pi)/(4)`

C

1

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
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lim_(n rarr oo)[(1)/(n^(2))+(2)/(n^(2))+....+(n)/(n^(2))]

lim_(n rarr oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))+...+(n)/(n^(2)))

Evaluate the following (i) lim_(n to oo)((1)/(n^(2))+(2)/(n^(2))+(3)/(n^(2))....+(n-1)/(n^(2))) (ii) lim_(n to oo)((1)/(n+1)+(1)/(n+2)+....+(1)/(2n)) (iii) lim_(n to oo)((n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+(n)/(2n^(2))) (iv) lim_(n to oo)((1^(p)+2^(p)+.....+n^(p)))/(n^(p+1)),pgt0

The value of lim_(n to oo)[(n)/(n^(2))+(n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+...+(n)/(n^(2)+(n-1)^(2))] is :

lim_(n to oo)[(n+1)/(n^(2)+1^(2))+(n+2)/(n^(2)+2^(2))+....+(1)/(n)]

The value of lim_(n rarr oo)[(n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))++(1)/(2n)] is

lim_(n to oo ) {(n)/(n^(2)+1^(2))+(n)/(n^(2)+2^(2))+....+ (n)/(n^(2)+n^(2))} is equal to

lim_(n rarr oo)(e^(2n)(n!)^(2))/(2n^(2n+1))

MARVEL PUBLICATION-INTEGRATION - DEFINITE INTEGRALS -MULTIPLE CHOICE QUESTIONS (PART - B : Mastering The BEST)
  1. int(1/n)^((an-1)/n) sqrt(x)/(sqrt(a-x)+sqrt(x))dx= (A) a/2 (B) (na+2)/...

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  2. Let a, b and c be non - zero real numbers such that int (0)^(3) (3ax...

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  3. lim(n to oo)[(n)/(1+n^(2))+(n)/(4+n^(2))+(n)/(9+n^(2))+…+(1)/(2n^2)]=

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  4. Evaluate lim(n->oo) [1^2/(n^3+1^3) + 2^2/(n^3+2^3) +3^2/(n^3+3^3) +......

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  5. lim(n->oo)1/nsum(r=1)^(2n)r/(sqrt(n^2+r^2)) equals

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  6. lim(n->oo) (1^p+2^p+3^p+...........+n^p)/n^(p+1)

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  7. The solution for x of the equation int(sqrt(2))^x(dt)/(tsqrt(t^2-1))=p...

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  8. integrate int0^(2pi) e^x . sin (pi/4 + x/2) dx

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  9. Evaluate : int0^(pi/2)(2logsinx-logsin2x)\ dx

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  10. If int0^(pi)xf(sinx)dx=Aint0^(pi/2)f(sinx)dx, then A is

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  11. The integral underset(0)overset(a)int (g(x))/(f(x)+f(a-x))dx vanishes...

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  12. If I(n)=underset(0)overset(pi//4)inttan^(n)x dx, then (1)/(I(2)+I(4)...

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  13. The value int(-2)^(2) (plog ((1+x)/(1-x)) + q log ((1-x)/(1+x))^(-2) +...

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  14. lim(n to oo)((sum(r=1)^(n)r^(2))(sum(r=1)^(n)r^(3)))/((sum(r=1)^(n)r^(...

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  15. If : int(0)^(pi)ln(sin x) dx = k, then : int(0)^(pi//4)ln (1+tan x)=

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  16. int(pi/3)^(pi/2) sqrt(1+cosx)/(1-cosx)^(5/2) dx

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  17. int(-pi//2)^(pi//2)(sin^(2)x.cos^(2)x(sin x+cos x))dx=

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  18. If : int(ln 2)^(x)(1)/(sqrt(e^(t)-1))dt=(pi)/(6), then : x =

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  19. int(0)^(pi//2)sin x.cos x. sqrt(a^(2).sin^(2)x+b^(2).cos^(2)x)dx=

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  20. If int0^oo (x^2 \ dx)/((x^2+a^2)(x^2+b^2)(x^2+c^2)) = pi/(2(a+b)(b+c)(...

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