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lim(nrarroo) sum(k=1)^(n)(k^(1//a{n^(a-(...

`lim_(nrarroo) sum_(k=1)^(n)(k^(1//a{n^(a-(1)/(a))+k^(a-(1)/(a))}))/(n^(a+1))` is equal to

A

1

B

2

C

43467

D

4

Text Solution

Verified by Experts

The correct Answer is:
A
`underset(nrarroo)(lim)sum_(k=1)^(n)(k^(1//a){n^(a-(1)/(a))+k^(a-(1)/(a))})/(n^(a+1))`
`" "=underset(nrarroo)(lim)sum_(k=1)^(n)(1)/(n).{((k)/(n))^(1//a)+((k)/(n))^(a)}`
`" "=int_(0)^(1)(x^(1//a)+x^(a))dx`
`" "={(x^((1//a)+1))/((1)/(a)+1)+(x^(a+1))/(a+1)}_(0)^(1)`
`" "=(a)/(a+1)+(1)/(a+1)=1`
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