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lim(nrarroo) [(1)/(n)+(n^(2))/((n+1)^(3)...

`lim_(nrarroo) [(1)/(n)+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+...+(1)/(8n)]` is equal to

A

`(3)/(8)`

B

`(1)/(4)`

C

`(1)/(8)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(nrarroo)(lim)[(1)/(n)+(n^(2))/((n+1)^(3))+(n^(2))/((n+2)^(3))+...+(1)/(8n)]`
`" "=underset(nrarroo)(lim)[(n^(2))/((n+0)^(3))+(n^(2))/((n+2)^(3))+(n^(2))/((n+2)^(3))+...+(n^(2))/((n+n)^(3))]`
`" "=underset(nrarroo)(lim)sum_(r=0)^(n)(n^(2))/((n+r)^(3))`
`" "=underset(nrarroo)(lim)sum_(r=0)^(n)(1)/(n)(1)/((1+(r)/(n)))`
`" "=int_(0)^(1)(dx)/((1+x^(3)))=[-(1)/(2(1+x)^(2))]_(0)^(1)`
`" "=-(1)/(2)((1)/(4)-1)=(3)/(8)`
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