Express limit as definite integral?

Question

limit as n goes to infinity.

 

n

Σ     i⁴/n⁵     on interval [0,1]

i=1

 

Options are: 

1             1              1            7          1

∫  1/x⁵  , ∫  1/x^{4    ,} ∫  x^{4   ,}   ∫  x^{5    ,}  ∫  x⁵/5

0             0               0           0          0

 

Please help!

Answer 1

You can rewrite the sum as 

 

(1/n) ∑[i = 1, n] (i/n)⁴

 

which is a Riemann sum for the continuous function f(x) = x⁴ over the interval [0,1]. So the limit of your sequence is 

 

∫[0,1] x⁴ dx.