Exercise 3, § 3

Functions of a Real Variable, VI, 297

Let $f$ be a continuous function with $2p + 1$ continuous derivatives on the interval $[0, 1]$ . Establish the formula

\int_0^1 f(t)\,dt = \frac{1}{n} \left( \frac{1}{2} f(0) + f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \cdots + f\left(\frac{n-1}{n}\right) + \frac{1}{2} f(1) \right)

- \sum_{k=1}^{2p} \frac{b_{2k}}{(2k)!} \frac{1}{n^{2k}} (f^{(2k-1)}(1) - f^{(2k-1)}(0)) + R_p(n)

with

|R_p(n)| \leq \frac{4e^{2\pi}}{(2\pi)^{2p+1}} \frac{1}{n^{2p+1}} \int_0^1 |f^{(2p+1)}(t)|\,dt.

Solution

No solution has been written yet.