73-74 express the limit as a definite integral. 73. \lim _(n \rightarrow \infty) \sum_(i=1)^(n) (i^(4))/(n^(5)) [hint: consider $f\mleft(x\mright)=x^4$.\rbrack

Question

73-74 express the limit as a definite integral.

73.
`\lim _(n \rightarrow \infty) \sum_(i=1)^(n) (i^(4))/(n^(5))` [hint: consider
`$f\mleft(x\mright)=x^4$.\rbrack`

Answer 1

Pull out a factor of 1/n from the summand and the result follows.

`\displaystyle \lim_(n\to\infty) \sum_(i=1)^n (i^4)/(n^5) = \lim_(n\to\infty) \frac1n \sum_(i=1)^n \left(\frac in\right)^4 = \boxed{\int_0^1 x^4 \, dx}`