Let $ f(x) = x $ and $ P = \left\{ 0, \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n}{n} = 1 \right\} $.Show that \[L_f(P) = f(0)\frac{1}{n} + f\left(\frac{1}{n}\right)\frac{1}{n} + \cdots + f\left(\frac{n-1}{n}\right)\frac{1}{n}\]and that \[\lim_{n \to \infty} L_f(P) = \frac{1}{2}\](Hint: You can use the identity $ 1 + 2 + \cdots + k = k(k + 1)/2 $)

Given the function f(x) = x & a partition of the interval 0,1 as P =0,1n,2n,3n,...nn=1. First we…

Answered: 3. Let f(x) = x and P = {0, nn = 1}.…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let $ f(x) = x $ and $ P = \left\{ 0, \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n}{n} = 1 \right\} $.

Show that

\[
L_f(P) = f(0)\frac{1}{n} + f\left(\frac{1}{n}\right)\frac{1}{n} + \cdots + f\left(\frac{n-1}{n}\right)\frac{1}{n}
\]

and that

\[
\lim_{n \to \infty} L_f(P) = \frac{1}{2}
\]

(Hint: You can use the identity $ 1 + 2 + \cdots + k = k(k + 1)/2 $)

Expert Solution

Step 1

Given the function f(x) = x & a partition of the interval $[0, 1]$ as $P = \{0, \frac{1}{n}, \frac{2}{n}, \frac{3}{n}, . . . \frac{n}{n} = 1\}$ .

First we have to show to show that $L_{f} (P) = f (0) \frac{1}{n} + f (\frac{1}{n}) \frac{1}{n} + . . . + f (\frac{n - 1}{n}) \frac{1}{n}$

We have the formula for $L_{f} (P)$ as follows,

$L_{f} (P) = \sum_{i = 1}^{n} m_{i} ∆ x_{i}$ , where $∆ x_{i}$ is the length of the $i^{t h}$ interval & $m_{i} = i n f \{f (x) : x_{i - 1} \leq x \leq x_{i}\}$ .

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