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

Find `a` for which `lim_(n->oo) (1^a+2^a+3^a+...+n^a)/((n+1)^(a-1)[(na+1)+(na+2)+...+(na+n)])=1/60`

A

5

B

7

C

`(-15)/2`

D

`(-17)/2`

Text Solution

Verified by Experts

The correct Answer is:
B, D
Given limit `=(lim_(n to oo) 1/n sum_(r=1)^(n)(r/n)^(a))/(lim_(n to oo) (1+1/n)^(a-1)lim_(n to oo) 1/n sum_(r=1)^(n)(a+r/n))=(int_(0)^(1)x^(a)dx)/(int_(0)^(1)(a+x)dx)`
`=2/((2a+1)(a+1))=2/120`
`:.a=7` or `-17/2`
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