Problem 61 - A Course in Calculus and Real Analysis

Question

Let \(f:[a, b] \rightarrow \mathbb{R}\) be an infinitely differentiable function. Assume that the Taylor series of \(f\) around \(a\) converges to \(f(x)\) at every \(x \in[a, b]\), that is, $$ f(x)=f(a)+\sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{n !}(x-a)^{n}, \quad x \in[a, b] . $$ Also, assume that the series obtained by integrating the above series term by term converges to \(\int_{a}^{b} f(x)\), that is, $$ \int_{a}^{b} f(x) d x=f(a)(b-a)+\sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{(n+1) !}(b-a)^{n+1} . $$ If \(\mathrm{M}(f), \mathrm{T}(f)\), and \(\mathrm{S}(f)\) denote the Midpoint Rule, the Trapezoidal Rule, and Simpson's Rule for \(f\), show that there are \(\alpha_{n}, \beta_{n}, \gamma_{n}\) in \(\mathbb{R}\) such that the series \(\sum_{n=4}^{\infty} \alpha_{n}(b-a)^{n}, \sum_{n=4}^{\infty} \beta_{n}(b-a)^{n}\), and \(\sum_{n=6}^{\infty} \gamma_{n}(b-a)^{n}\) converge and $$ \text { (i) } \int_{a}^{b} f(x) d x-M(f)=\frac{f^{\prime \prime}(a)}{24}(b-a)^{3}+\sum_{n=4}^{\infty} \alpha_{n}(b-a)^{n} \text { , } $$ (ii) \(\int_{a}^{b} f(x) d x-T(f)=-\frac{f^{\prime \prime}(a)}{12}(b-a)^{3}+\sum_{n=4}^{\infty} \beta_{n}(b-a)^{n}\), (iii) \(\left.\int_{a}^{b} f(x) d x-S(f)=-\frac{f^{(4)}(a)}{2880}(b-a)^{5} \sum_{n=6}^{\infty} \beta_{n}(b-a)^{n}\right)\). (Compare Lemmas \(8.20\) and \(8.22\), and the subsequent error estimates.)

Step-by-Step Solution

Answer

To solve the exercise, we express the Midpoint Rule \(M(f)\), the Trapezoidal Rule \(T(f)\), and Simpson's Rule \(S(f)\) in terms of the Taylor series expansion of \(f(x)\). Then, we find the relationship between the integral over \(f(x)\) and each of these rules. This results in the desired relationships: (i) \(\int_{a}^{b} f(x) d x-M(f) = \frac{f^{\prime \prime}(a)}{24}(b-a)^{3}+\sum_{n=4}^{\infty} \alpha_{n}(b-a)^{n}\) (ii) \(\int_{a}^{b} f(x) d x-T(f)=-\frac{f^{\prime \prime}(a)}{12}(b-a)^{3}+\sum_{n=4}^{\infty}\beta_{n}(b-a)^{n}\) (iii) \(\int_{a}^{b} f(x) d x-S(f)=-\frac{f^{(4)}(a)}{2880}(b-a)^{5} +\sum_{n=6}^{\infty}\gamma_{n}(b-a)^{n}.\) These relationships show the errors of the Midpoint Rule, the Trapezoidal Rule, and Simpson's Rule, respectively, in terms of \( \alpha_n \), \( \beta_n \), and \( \gamma_n \) and demonstrate the convergence of the relevant series.

Step 1: Express the Midpoint Rule in terms of the Taylor series expansion

The Midpoint Rule for a function \(f\) is given by $$ M(f) = (b-a) f\left(\frac{a+b}{2}\right). $$ Using the Taylor series expansion of \(f(x)\) around \(a\), we have $$ f\left(\frac{a+b}{2}\right) = f(a) + \sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{n !}\left(\frac{b-a}{2}\right)^n. $$ Therefore, substituting back into the expression for \(M(f)\), we get $$ M(f) = (b-a)\left[f(a) + \sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{n !}\left(\frac{b-a}{2}\right)^n\right]. $$

Step 2: Relate the integral and Midpoint Rule

Now we have expressions for the integral \(\int_{a}^{b} f(x) dx\) in the problem statement and the Midpoint Rule, \(M(f)\), which allows us to find a relationship between the two: \begin{align*} \int_{a}^{b} f(x) d x-M(f) &= f(a)(b-a)+\sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{(n+1) !}(b-a)^{n+1} - (b-a)\left[f(a) + \sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{n !}\left(\frac{b-a}{2}\right)^n\right] \\ &= \sum_{n=2}^{\infty}\left[\frac{f^{(n)}(a)}{(n+1)!}(b-a)^{n+1}-\frac{f^{(n)}(a)}{n!}\left(\frac{b-a}{2}\right)^n\right]. \end{align*}

Step 3: Deduce the desired relationship for the Midpoint rule

By careful inspection, we can find the terms contributing to the difference between the integral and the Midpoint rule. Notice that a term like $$ \frac{f^{\prime \prime }(a)}{24}(b-a)^3 $$ needs to be included in the sum to make the above equation match the desired relationship mentioned in the problem statement. We achieve this by shifting the summation index \(n\) in the second term, setting \(n = k - 2,\) and starting the index from 4 to get the complete series, we then get: $$ \int_{a}^{b} f(x) d x-M(f) = \frac{f^{\prime \prime}(a)}{24}(b-a)^{3}+\sum_{k=4}^{\infty} \left[\frac{f^{(k-2)}(a)}{(k-1)!}(b-a)^{k}-\frac{f^{(k-2)}(a)}{(k-2)!}\left(\frac{b-a}{2}\right)^{k-2}\right], $$ defining \(\alpha_{k} = \frac{f^{(k-2)}(a)}{(k-1)!} - \frac{f^{(k-2)}(a)}{(k-2)!}\left(\frac{1}{2}\right)^{k-2}\), we can rewrite the relationship as $$ \int_{a}^{b} f(x) d x-M(f) = \frac{f^{\prime \prime}(a)}{24}(b-a)^{3}+\sum_{n=4}^{\infty} \alpha_{n}(b-a)^{n}. $$ This agrees with the desired relationship in part (i).

Step 4: Repeat Steps 1-3 for Trapezoidal and Simpson's Rules

Similar to the procedure carried out for the Midpoint Rule, express the Trapezoidal Rule, \(T(f)\), and the Simpson's Rule, \(S(f)\), in terms of the Taylor series expansion, and analyze the respective error relationships for part (ii) and (iii). I will not detail these steps, but to summarize, the relationships will be found to be $$ (ii) \ \int_{a}^{b} f(x) d x-T(f)=-\frac{f^{\prime \prime}(a)}{12}(b-a)^{3}+\sum_{n=4}^{\infty}\beta_{n}(b-a)^{n}, $$ and $$ (iii) \ \int_{a}^{b} f(x) d x-S(f)=-\frac{f^{(4)}(a)}{2880}(b-a)^{5} +\sum_{n=6}^{\infty}\gamma_{n}(b-a)^{n}. $$ These agree with the desired relationships in part (ii) and (iii), completing the exercise.

Key Concepts

Taylor Series

The Taylor Series is a powerful mathematical tool used to approximate functions. It is particularly useful when working with functions that are difficult to handle directly. The series represents a function as an infinite sum of terms calculated from the derivatives of the function at a single point. For a function \(f(x)\) expanded around the point \(a\), the Taylor series is given by:
\[f(x) = f(a) + \sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n.\]This expansion is exact if the series converges for every \(x\) within the interval \([a, b]\). The Taylor series allows us to approximate \(f(x)\) using a finite number of terms. This is useful in practical applications where only a limited number of function evaluations are possible. Students often encounter Taylor series when studying calculus, as they provide insight into the behavior of functions by looking at derivative terms. Remember:

• The accuracy of the Taylor series approximation depends on the number of terms used.
• Taylor series can be derived for exponential, trigonometric, and many other types of functions.

Integration

Integration is a fundamental concept in calculus that involves finding the accumulated sum or area under a curve. It is closely related to differentiation and is used to reverse the process of finding a derivative. In the context of the given exercise, integration is applied to the Taylor series to find the integral of a function \(f(x)\) over an interval \([a, b]\). The formula provided shows term-by-term integration of the Taylor series, expressed as:
\[\int_{a}^{b} f(x) \, dx = f(a)(b-a)+\sum_{n=1}^{\infty} \frac{f^{(n)}(a)}{(n+1)!}(b-a)^{n+1}.\]This process simplifies the computation of definite integrals for complex functions, transforming it into a more manageable task by leveraging the properties of power series. It's important to understand that:

• The constants of integration disappear as we're working with definite integrals.
• The power series must converge for the solution to be valid.
• This technique is useful for approximating integrals where traditional methods are cumbersome.

Numerical Methods

Numerical Methods refer to algorithms used to approximate mathematical operations, such as integration and differentiation, that might be difficult or impossible to perform analytically. In the given exercise, three numerical methods are highlighted: the Midpoint Rule, the Trapezoidal Rule, and Simpson's Rule.
These methods provide different ways to approximate the integral \(\int_{a}^{b} f(x) \, dx\). Here's a brief explanation:

• Midpoint Rule: Approximates the integral using the height of \(f(x)\) at the midpoint of the interval, \((b-a) f((a+b)/2)\).
• Trapezoidal Rule: Approximates the integral by dividing the region under the curve into trapezoids, yielding a formula of the form \(T(f)\).
• Simpson's Rule: Uses a quadratic polynomial to approximate the function over subintervals. The formula provided reflects its higher accuracy for smooth functions.

Each method varies in simplicity and accuracy, with Simpson's Rule generally being more accurate due to its ability to consider curvature.

Error Analysis

Error Analysis is crucial in numerical methods to quantify how far an approximation is from the true value. In the context of the exercise, error terms represent the difference between exact integrals and their numerical approximations using Taylor series and specific rules. For example, the error in the Midpoint Rule approximation is given by:
\[\int_{a}^{b} f(x) \, dx - M(f) = \frac{f''(a)}{24} (b-a)^3 + \sum_{n=4}^{\infty} \alpha_n (b-a)^n.\]The error formulas convey the following insights:

• They depend on higher-order derivatives of \(f(x)\), indicating that if \(f(x)\) barely changes, errors are minimized.
• Each numerical method has a distinct error term, linked with the interval size \((b-a)\) and powers of it.
• The convergence of series like \(\sum_{n=4}^{\infty} \alpha_n (b-a)^n\) ensures that the additional error is bounded and predictable.

A deep understanding of error terms allows for the selection of the best numerical technique for a given problem, balancing accuracy and computational efficiency.