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lim(n to oo) {(1)/(n)+(1)/(n+1)+(1)/(n+2...

`lim_(n to oo) {(1)/(n)+(1)/(n+1)+(1)/(n+2)+...+(1)/(3n)}=`

A

log2

B

log3

C

log5

D

0

Text Solution

Verified by Experts

The correct Answer is:
B
`S=underset(n to oo)lim{(1)/(n)+(1)/(n+1)+(1)/(n+2)+...+(1)/(3n)}`
`S=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(1)/(n+r)}`
`S=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(n)/(n+r)}=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(1)/(1+(r//n))}`
Now, Lower limit `underset(nto oo)lim(r)/(n)`
`underset(nto oo)lim(r)/(n)=0" " [ :. r=0 "for the first term"]`
Upper Limit `underset(nto 0)lim(r)/(n)`
`underset(nto oo)lim(r)/(n)" " [ :. r=2n "for the last term"]`
`:. S=underset(n to oo)lim{(1)/(n)underset(r=0)overset(2n)sum(1)/(1+r(//n))}`
`rArr S=underset(0)overset(2)int(1)/(1+x)dx=[ log(1+x)]_(0)^(2)=log3-log=log3`
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OBJECTIVE RD SHARMA ENGLISH-DEFINITE INTEGRALS-Chapter Test 2
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  3. The value of integral sum (k=1)^(n) int (0)^(1) f(k - 1+x) dx is

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  4. Let f (x) be a function satisfying f(x)=f(x) with f(0) = 1 and g be th...

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  5. If I=int(0)^(1)cos(2 cot^(-1)sqrt(((1-x)/(1+x))))dx then :

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  6. The value of int(a)^(a+(pi//2))(sin^(4)x+cos^(4)x)dx is

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  7. The vaue of int(-1)^(2) (|x|)/(x)dx is

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  8. The value of int0^1 (x^(3))/(1+x^(8))dx is

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  9. The value of int(0)^(3) xsqrt(1+x)dx, is

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  10. Evaluate int(0)^(1)log(sin((pix)/(2)))dx

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  11. Evaluate int(0)^(pi) xlog sinx dx

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  12. If I(1)=int(0)^(oo) (dx)/(1+x^(4))dx and I(2)=underset(0)overset(oo)i...

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  13. If f(x)={{:(x,xlt1),(x-1,xge1):}, then underset(0)overset(2)intx^(2)f(...

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  14. The value of the integral overset(1)underset(0)int (1)/((1+x^(2))^(3//...

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  15. Prove that: int0^(2a)f(x)dx=int0^(2a)f(2a-x)dxdot

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  16. If int(0)^(36) (1)/(2x+9)dx =log k, is equal to

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  17. The value of the integral int(0)^(pi//2) sin^(6) x dx, is

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  18. If int(0)^(oo) e^(-x^(2))dx=sqrt((pi)/(2))"then"int(0)^(oo) e^(-ax^(2)...

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  19. The value of the integral int 0^oo 1/(1+x^4)dx is

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  20. The value of alpha in [0,2pi] which does not satify the equation int(p...

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  21. lim(x to 0)(int(0)^(x^(2))sinsqrt(t) dt)/(x^(3)) is equl to

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