Question

Express the following limits in the form of an integral. $$ \begin{aligned} &\text { (i) } \lim _{\mathbb{a} \rightarrow \infty} \frac{1}{\mathrm{n}}\left(\cos \frac{\pi}{\mathrm{n}}+\cos \frac{2 \pi}{\mathrm{n}}+\ldots+\cos \frac{\mathrm{n} \pi}{\mathrm{n}}\right) \\ &\text { (ii) } \lim _{\mathrm{n} \rightarrow x} \frac{\pi}{6 \mathrm{n}}\left[\sec ^{2}\left(\frac{\pi}{6 \mathrm{n}}\right)+\sec ^{2}\left(2 \frac{\pi}{6 \mathrm{n}}\right)+\ldots\right. \\ &\left.\ldots+\sec ^{2}\left((\mathrm{n}-1) \frac{\pi}{6 \mathrm{n}}\right)+\frac{4}{3}\right] \end{aligned} $$

Short answer

- The function to be integrated is \(f(x)=\cos(\pi x)\). 2) Over which interval should the function in part (i) be integrated? - The function should be integrated over the interval [0, 1]. 3) What function should be integrated to represent the limit in part (ii)? - The function to be integrated is \(f(x)=\sec^2(\frac{\pi}{6}x)\). 4) Over which interval should the function in part (ii) be integrated? - The function should be integrated over the interval [0, 1]. 5) What constant should be added to the integral in part (ii) to represent the given limit? - 4/3 should be added to the integral.

Step-by-step solution

(i) Change the limit to Riemann sum

To find the integral representation of the given limit, we need to identify the function and its corresponding interval in the limit. The limit takes the following form: $$ \lim_{n \rightarrow \infty} \frac{1}{n} \left(\cos \frac{\pi}{n}+\cos \frac{2 \pi}{n}+\ldots+\cos \frac{n \pi}{n}\right) $$ We can rewrite it in the Riemann sum form, with the function \(f(x) = \cos(\pi x)\), and \(x_k = \frac{k}{n}\) for \(k=1,2,\dots,n\): $$ \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n f(x_k) $$

(i) Find the integral representation

Using the Riemann sum definition of the integral, the limit can be represented by the integral of the function \(f(x)=\cos(\pi x)\) over the interval \([0, 1]\). So, we have: $$ \lim_{n \rightarrow \infty} \frac{1}{n}\left(\cos \frac{\pi}{n}+\cos \frac{2 \pi}{n}+\ldots+\cos \frac{n \pi}{n}\right) = \int_0^1 \cos (\pi x) dx $$

(ii) Change the limit to Riemann sum

Similar to part (i), we need to identify the function and the corresponding interval in the limit. The limit takes the form: $$ \lim_{n \rightarrow \infty} \frac{\pi}{6n} \left[\sec^2\left(\frac{\pi}{6n}\right)+\sec^2\left(2 \frac{\pi}{6n}\right)+\ldots+\sec^2\left((n-1) \frac{\pi}{6n}\right)+\frac{4}{3}\right] $$ We can rewrite it in the Riemann sum form, with the function \(f(x) = \sec^2(\frac{\pi}{6}x)\) (note that we need to subtract the constant term \(\frac{4}{3}\) after we evaluate the integral), and \(x_k = \frac{k}{n}\) for \(k=1,2,\dots,n-1\): $$ \lim_{n \rightarrow \infty} \frac{\pi}{6n} \left(\sum_{k=1}^{n-1} f(x_k)+\frac{4}{3}\right) $$

(ii) Find the integral representation

Now, we represent the limit using the integral of the function \(f(x)=\sec^2(\frac{\pi}{6}x)\) over the interval \([0,1]\). Don't forget to include the constant \(\frac{4}{3}\) after evaluating the integral: $$ \lim_{n \rightarrow \infty} \frac{\pi}{6n} \left[\sec^2\left(\frac{\pi}{6n}\right)+\sec^2\left(2 \frac{\pi}{6n}\right)+\ldots+\sec^2\left((n-1) \frac{\pi}{6n}\right)+\frac{4}{3}\right] = \int_0^1 \sec^2\left(\frac{\pi}{6}x\right) dx + \frac{4}{3} $$