Finding Limits 6

Calculus Level pending

$\large \lim_{n \to \infty} \left( \dfrac{1^p}{n^{p+1}} + \dfrac{2^p}{n^{p+1}} + \dfrac{3^p}{n^{p+1}} + \cdots + \dfrac{n^p}{n^{p+1}} \right) = \, ?$

Evaluate the limit above for constant $p>-1$ .

1

p+1

\infty

\frac{1}{p+1}

p

0

1 solution

Chew-Seong Cheong
Aug 17, 2016

Relevant wiki: Riemann Sums

$\begin{aligned} L & = \lim_{n \to \infty} \sum_{k=1}^n \frac {k^p}{n^{p+1}} \\ & = \lim_{n \to \infty} \frac 1n \sum_{k=1}^n \left( \frac kn \right)^p & \small \color{#3D99F6}{\text{By Riemann sums}} \\ & = \int_a^b x^p dx & \small \color{#3D99F6}{a = \lim_{n \to \infty} \frac 1n = 0, \ b = \lim_{n \to \infty} \frac nn = 1} \\ & = \int_0^1 x^p dx \\ & = \boxed{\dfrac 1{p+1}} \end{aligned}$

Nice solution. Thank you.

Hana Wehbi - 4 years, 10 months ago

You are welcome.

Chew-Seong Cheong - 4 years, 10 months ago

Finding Limits 6

1 solution

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