By the definition, for sure!

Calculus Level 2

lim ⁡ n → ∞ [ ( r + 1 ) ∑ k = 0 n − 1 k r n r + 1 ] \large \displaystyle \lim_{n\to\infty} \left [ (r+1) \frac{\displaystyle \sum_{k=0}^{n-1} k^r}{n^{r+1}} \right ]

For positive r r , evaluate the limit above.


The answer is 1.

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Mikael Marcondes by Mikael Marcondes , 24, Brazil

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1 solution

Otto Bretscher
Sep 27, 2015

The expression whose limit we take is the lower (or left) Riemann sum of f ( x ) = ( r + 1 ) x r f(x)=(r+1)x^r on the interval [ 0 , 1 ] [0,1] , with n n subintervals of equal length Δ x = 1 n \Delta{x}=\frac{1}{n} . Thus the limit is ∫ 0 1 f ( x ) d x = 1 \int_{0}^{1}f(x)dx=1 .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Exactly how I wanted people to solve it. Nice done, sir.

Mikael Marcondes - 5 years, 8 months ago

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