Finding Limits 5

Calculus Level 3

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

0

\frac{\pi}{4}

\pi

\infty

1 solution

Chew-Seong Cheong
Aug 17, 2016

Relevant wiki: Riemann Sums

$\begin{aligned} L & = \lim_{n \to \infty} \sum_{k=1}^n \frac n{n^2+k^2} & \small \color{#3D99F6}{\text{Divide up and down by }n} \\ & = \lim_{n \to \infty} \sum_{k=1}^n \frac 1{n+\frac {k^2}n} \\ & = \lim_{n \to \infty} \frac 1n \sum_{k=1}^n \frac 1{1+\left(\frac kn\right)^2} & \small \color{#3D99F6}{\text{By Riemann sums}} \\ & = \int_a^b \frac 1{1+x^2} dx & \small \color{#3D99F6}{a = \lim_{n \to \infty} \frac 1n = 0, \ b = \lim_{n \to \infty} \frac nn = 1} \\ & = \int_0^1 \frac 1{1+x^2} dx \\ & = \tan^{-1} x \ \bigg|_0^1 \\ & = \boxed{\dfrac \pi 4} \end{aligned}$

Thank you again for the elegant and clear solution :)

Hana Wehbi - 4 years, 10 months ago

Nice to know that you like it. Thanks for the problem.

Chew-Seong Cheong - 4 years, 10 months ago

Finding Limits 5

1 solution

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