Exhaustive Study Notes on Infinite Series and Taylor Polynomials

Introduction to Infinite Series and Polynomial Approximations

• Conceptual Overview: This week focuses on infinite series that involve polynomial terms. These are essentially polynomials with infinitely many terms.

• Primary Application: The core idea is to use polynomials to approximate functions that are not polynomials themselves. The primary example of this is the Taylor series.

• Objective: To determine if a polynomial $p_n(x)$ can be found such that it is approximately equal to a given function $f(x)$.

Fundamental Properties of Polynomials

• Notation: An $n$-th degree polynomial in one variable (where $x$ is the input) is commonly denoted as $p_n(x)$.

• Coefficients and Explicit Rules: A polynomial can be expressed algebraically as:     $p_n(x) = p_n x^n + p_{n-1} x^{n-1} + \dots + p_1 x + p_0$     * The terms $p_0, p_1, \dots, p_n$ are the coefficients of the polynomial, which are assumed to be real numbers.

• The Degree: The degree of a polynomial is determined by the highest degree term. For a polynomial to be strictly called an $n$-th degree polynomial, the leading coefficient ($p_n$) must be nonzero ($p_n \neq 0$).

• Natural Number Requirement: For a function to qualify as a polynomial, the power ($n$) must be a natural number (a positive whole number).     * Functions with negative powers are not polynomials.     * Functions with rational powers (e.g., $x^{1/2}$) are not polynomials.

Motivation: Why Approximate with Polynomials?

• The Best Fit Principle: If the function $f$ is already a polynomial, then the polynomial of best fit already exists and is simply $f$ itself. In some specialized cases (e.g., in other classes), one might seek a lower-degree polynomial to see which terms impact the function most significantly.

• Complexity of Non-Polynomial Functions: Functions such as $\sin(x)$, $\tan(x)$, $\frac{1}{x}$, and $e^x$ are harder to evaluate exactly. For example, $\tan(x)$ has vertical asymptotes, and evaluating $\cos(x)$ at arbitrary points is generally challenging.

• Polynomials as "Ideal" Functions: Polynomials are described as being "nice" and easy to work with for Plusieurs reasons:     * Arithmetic Simplicity: They only require basic arithmetic: addition and multiplication. Exponentiation to a whole number is just a finite product (e.g., $x^2 = x \times x$).     * Domain and Continuity: Polynomials are defined over the entire set of real numbers ($\mathbb{R}$). They are continuous, meaning they have no breaks, asymptotes, jumps, or holes.     * Calculus Properties: They are both differentiable and integrable over their entire domain.     * Computational Efficiency: They are easier to compute than transcendental functions, both for humans and for calculators.

Review of Linearization (First-Degree Polynomial Approximation)

• Linearization from Math 1A: A low-degree polynomial approximation of $f(x)$. It is specifically a first-degree approximation passing through a point $(a, f(a))$.

• Geometric Formula: The linearization corresponds to the tangent line through point $(a, f(a))$:     $L(x) = f'(a)(x - a) + f(a)$     * $f'(a)$: Representing the slope of the tangent line.     * $(x - a)$: The horizontal distance from the center point.

• Local Accuracy: If you zoom in close enough to the center point $a$, the tangent line is indistinguishable from the curve of function $f$. Outside this local interval, the approximation typically becomes poor.

• Prerequisites: For a linearization to exist, the function $f$ must be differentiable at the point $a$. This implies $a$ must be in the domain of $f$.

The Remainder and Higher-Degree Terms

• Definition of Remainder: The error of an approximation is the difference between the true function value and the approximate value:     $\text{Remainder} = \text{True Value} - \text{Approximate Value}$

• Notation for First-Degree Remainder: $R_1(x) = f(x) - L(x)$.

• Improving Accuracy: Generally, increasing the degree ($n$) of the polynomial allows the inclusion of more terms, yielding a better approximation.

• Centering: To construct an approximation, one must fix a center point $a$. At this specific point, the remainder is guaranteed to be zero: $p_n(a) = f(a)$.

Constructing Taylor Polynomials: Formal Requirements

• Requirements for Alignment: The goal is to find coefficients such that the Taylor polynomial and all its derivatives align with the function and its derivatives at the center point $a$.     $T_n^{(i)}(a) = f^{(i)}(a)$     * Where $i$ ranges from $0$ (the function itself) to $n$ (the degree of the polynomial).

• Polynomial Differentiation Properties: The $(n+1)$-th derivative of an $n$-th degree polynomial is zero. Therefore, the $n$-th derivative is a constant.

• Deriving the Constant term via FTC: By using the Fundamental Theorem of Calculus (FTC) and anti-differentiating, one can solve for the constants $c_i$. Matching the constant $n$-th derivative to $f^{(n)}(a)$ allows the build-up of the formula.

• Origin of the Factorial: When differentiating a polynomial term like $x^n$ repeatedly, the power rule brings down exponents ($n$, then $n-1$, etc.). By the time the $n$-th derivative is reached, we are left with a constant multiplied by $n!$. Consequently, the coefficients in the Taylor series involve a division by $i!$ to account for this growth.

The Taylor and Maclaurin Formulas

• The General Formula: The $n$-th degree Taylor polynomial centered at $x = a$ is:     $T_n(x) = \sum_{i=0}^n \frac{f^{(i)}(a)}{i!}(x - a)^i$

• Expansion: $T_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \dots + \frac{f^{(n)}(a)}{n!}(x - a)^n$

• Maclaurin Series: If the series is centered at zero ($a = 0$), it is often specifically called a Maclaurin series.

Example 1: The Exponential Function $e^x$

• Function Characteristics: The function $f(x) = e^x$ is infinitely differentiable. Its derivative is always itself ($f^{(i)}(x) = e^x$).

• Centered at Zero: Evaluating at $a = 0$ results in $f^{(i)}(0) = e^0 = 1$ for all $i$.

• N-th Degree Polynomial: Substituting these into the formula (where $0! = 1$):     $T_n(x) = \sum_{i=0}^n \frac{x^i}{i!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots + \frac{x^n}{n!}$

• Visualizing Improvement: The 0-degree approximation is a horizontal line ($y = 1$). The 1st-degree (linearization) adds a slope. The 2nd-degree adds curvature/concavity from the second derivative. As $n \rightarrow \infty$, the series converges toward the actual function $e^x$.

Example 2: The Rational Function $f(x) = \frac{1}{1 - x}$

• Setup: Centered at $a = 0$. Note that at $x = 1$, there is a vertical asymptote.

• Derivative Analysis:     * $f(x) = (1 - x)^{-1} \implies f(0) = 1$     * $f'(x) = 1(1 - x)^{-2} \implies f'(0) = 1$     * $f''(x) = 2(1 - x)^{-3} \implies f''(0) = 2 = 2!$     * $f'''(x) = 6(1 - x)^{-4} \implies f'''(0) = 6 = 3!$     * In general, the $i$-th derivative at zero is $f^{(i)}(0) = i!$.

• Polynomial Construction: Plugging into the general formula:     $T_n(x) = \sum_{i=0}^n \frac{i!}{i!} x^i = \sum_{i=0}^n x^i = 1 + x + x^2 + \dots + x^n$

• Connection to Geometric Series: This is a geometric series with a common ratio $r = x$. The infinite geometric series formula $\frac{a_0}{1 - r}$ yields $\frac{1}{1 - x}$, proving the Taylor series converges to the function on the interval where the derivatives exist and the ratio is within bounds.

Questions & Discussion

• Natural Numbers Concern: A student should review precalculus if they are unclear on natural numbers (positive whole numbers) for polynomial powers.

• Graphing Exercise: Students are encouraged to use graphing software to visualize the difference between $e^x$ and its various Taylor polynomial degrees ($T_0, T_1, T_2$).

• Future Lessons: Upcoming content will cover remainders and power series convergence, which formalize when these approximations exactly equal the source function.

• Note on Production: The speaker mentions this is the second time filming the video due to technical issues and records the session while feeling tired toward the end, resulting in a small initial error in drawing the rational function graph which was subsequently corrected.