Finding limit of Riemann Sum (1/n) sum[k=1,n] cos(4k pi / 3n)

[sktsasus]

Problem 5: Express the following expression:

. . . . .\(\displaystyle \displaystyle \dfrac{1}{n}\, \sum_{k=1}^n\, \cos\left(\dfrac{4k\pi}{3n}\right)\)

...as the Riemann sum for an integral of the form:

. . . . .\(\displaystyle \displaystyle \int_0^{4\pi/3}\, f(t)\, dt\)

...for a suitable function f.

Hence, find:

. . . . .\(\displaystyle \displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1}{n}\, \sum_{k=1}^n\, \cos\left(\dfrac{4k\pi}{3n}\right)\)

---

The main problem I am having here is that I am not sure how I would express that sum as a Riemann sum. I'm not entirely sure how we can use the constant k either. Am I supposed to use some identity or will it have to take some other form? I believe I am supposed to find the integral for f(t) between those two numbers but I am not exactly sure how I am supposed to get the equation for f(t).

Any help?

 

[barrick]

Problem 5: Express the following expression:

. . . . .\(\displaystyle \displaystyle \dfrac{1}{n}\, \sum_{k=1}^n\, \cos\left(\dfrac{4k\pi}{3n}\right)\)

...as the Riemann sum for an integral of the form:

. . . . .\(\displaystyle \displaystyle \int_0^{4\pi/3}\, f(t)\, dt\)

...for a suitable function f.

Hence, find:

. . . . .\(\displaystyle \displaystyle \lim_{n \rightarrow \infty}\, \dfrac{1}{n}\, \sum_{k=1}^n\, \cos\left(\dfrac{4k\pi}{3n}\right)\)

---

The main problem I am having here is that I am not sure how I would express that sum as a Riemann sum. I'm not entirely sure how we can use the constant k either. Am I supposed to use some identity or will it have to take some other form? I believe I am supposed to find the integral for [FONT=MathJax_Math-italic]f[FONT=MathJax_Main]([FONT=MathJax_Math-italic]t[FONT=MathJax_Main]) [/FONT][/FONT][/FONT]between those two numbers but I am not exactly sure how I am supposed to get the equation for [FONT=MathJax_Math-italic]f[FONT=MathJax_Main]([FONT=MathJax_Math-italic]t[FONT=MathJax_Main])[/FONT][/FONT][/FONT][/FONT].[/FONT]

Any help?

Hi sktsasus,

Here is a tip to get you started:

The integral is over the interval \(\displaystyle [0,4\pi/3]\), and the sum contains n terms. This suggests that you should divide the interval into n segments of length \(\displaystyle 4\pi/3n\) each.

You should then define \(\displaystyle f(x)\) in such a way that the sum corresponds to the Riemann sum of the integral.