University Mathematics II Compendium: From Infinite Series to Multivariable Calculus

Book Metadata and Author Information

Title: University Mathematics II (2017).
Target Courses: Applied Mathematics II, Calculus II, and Calculus of Functions of Several Variables.
Authors: Addisu W/Meskel (M.Sc.), Bizuneh Minda (M.Sc.), Getachew Bitew (M.Sc.), Temesgen Alemu (M.Sc.), and Tilahun Esayiyas (M.Sc.).
Affiliation: Lecturers of Mathematics at Addis Ababa University, Ethiopia.
Previous Work: University Mathematics I.
Publication Date: January, 2017.
Content Summary: The material contains two main parts: * Part I: Infinite Series: Chapters 1–3 covering Sequences and Series, Power Series, and Fourier Series. * Part II: Calculus of Functions of Several Variables: Chapters 4–7 covering Preliminaries, Limits and Continuity, Partial Derivatives and Applications, and Multiple Integrals.

Chapter 1: Sequences and Series

1.1 Definition of a Sequence

General Definition: A sequence is an ordered collection of objects or events with an identified first, second, and third member.
Mathematical Definition (1.1.1): A sequence is a function from the set of integers greater than or equal to some integer $m_0$ (usually 0 or 1) into a nonempty set $X$ .
Real Sequence: If the range is a subset of real numbers ( $R$ ), it is a sequence of real numbers. This chapter focuses exclusively on real sequences.
Notation: * General term: $a_n$ in place of $a(n)$ . * Terms: $a_1, a_2, a_3, \dots, a_n, \dots$ * Ordered pairs representation: ${(1, a_1), (2, a_2), \dots, (n, a_n), \dots}$ . * Shorthand: ${a_n}$ or ${a_n}_{n=1}^{\infty}$ .
Description Methods: 1. Listing: Writing the first few terms (e.g., ${2, 4, 6, 8, \dots}$ where $a_n = 2n$ ). 2. Defining Formula: Giving a specific rule for $a_n$ . For $a_n = \frac{2^n}{n-1}$ , the smallest value is $n=2$ . 3. Recursive Definition: Defining a term using preceding terms (recurrence). * Positive odd numbers: $a_1 = 1$ and $a_{n+1} = a_n + 2, \forall n \ge 1$ . * Fibonacci Sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$ s.t. $a_1 = a_2 = 1$ and $a_{n+2} = a_{n+1} + a_n, \forall n \ge 2$ . 4. Verbally: For cases with no obvious pattern, like the sequence of prime numbers.
Special Types: * Stationary Sequence: $a_n = a_{n+1}$ for all natural numbers $n$ . * Equality: ${a_n} = {b_n}$ if $a_n = b_n, \forall n$ .
Graph of a Sequence: Consists of isolated points on a coordinate plane with coordinates $(n_0, a_{n_0}), (n_1, a_{n_1}), \dots$ .

1.2 Convergence and Divergence of Sequences

Formal Definition (1.2.1): A sequence ${a_n}$ has the limit $L$ if for every \varepsilon > 0, there exists a positive integer $N$ (dependent on $\varepsilon$ ) such that for every n > N, |a_n - L| < \varepsilon. * Notation: $\lim_{n \to \infty} a_n = L$ or $a_n \to L$ as $n \to \infty$ . * A sequence is convergent if the limit exists; otherwise, it is divergent.
Geometric Representation: The points $(n, a_n)$ must lie between the horizontal lines $y = L + \varepsilon$ and $y = L - \varepsilon$ for all n > N.
Theorem 1.2.1: If $f(n) = a_n$ for a real-valued function $f$ , then $\lim_{x \to \infty} f(x) = L \implies \lim_{n \to \infty} a_n = L$ . * Caution: The converse is not always true. Example: $\cos(2\pi n) \to 1$ , but $\lim_{x \to \infty} \cos(2\pi x)$ does not exist.
Types of Divergent Sequences: * Diverge to $\infty$ : \forall M > 0, \exists N \text{ s.t. } n > N \implies a_n > M. * Diverge to $-\infty$ : \forall m > 0, \exists N \text{ s.t. } n > N \implies a_n < m. * Oscillating: Sequences that diverge but not to $\pm\infty$ .
Null Sequence: A sequence that converges to zero.
Geometric Sequences: The sequence ${cr^n}$ converges if -1 < r \le 1. * Limit is 0 if -1 < r < 1. * Limit is $c$ if $r = 1$ .

Chapter 2: Power Series

Definition (2.0.1): A power series in $(x-a)$ is a series of the form: $\sum_{n=0}^{\infty} c_n(x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + c_3(x-a)^3 + \dots$ where $x$ is a variable, $c_n$ are constants (coefficients), and $a$ is the center.
Radius and Interval of Convergence: * Theorem 2.1.1: Exactly one of the following is true: 1. Converges only at $x = a$ ( $R = 0$ ). 2. Absolutely convergent for all $x$ ( $R = \infty$ ). 3. \exists R > 0 such that series converges if |x-a| < R and diverges if |x-a| > R. * Abel’s Theorem (2.2.3): Relates the function limit to the series sum at endpoints of the interval of convergence.
Differentiation and Integration (2.1.4): A power series $P(x)$ with R > 0 is differentiable and integrable on $(a-R, a+R)$ term by term. The radius $R$ remains unchanged.
Uniqueness Theorem (2.2.1): If $f(x)$ is represented by a power series, the coefficients are $c_n = \frac{f^{(n)}(a)}{n!}$ .
Taylor Series: The power series representation $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$ . * Maclaurin Series: A Taylor series centered at $a = 0$ .
Common Series Representations: * $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ for all $x$ . * $\sin(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$ . * $\cos(x) = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$ . * $\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ for |x| < 1. * $\ln(1+x) = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}$ for -1 < x \le 1.
Binomial Series (2.4.1): For any real $m$ , $(1+x)^m = \sum_{n=0}^{\infty} \binom{m}{n} x^n$ where $\binom{m}{n} = \frac{m(m-1)\dots(m-n+1)}{n!}$ . Converges for |x| < 1.

Chapter 3: Fourier Series

Periodic Functions: $f(x + P) = f(x)$ . The smallest $P$ is the fundamental period.
Inner Product (3.1.4): $(f, g) = \int_a^b f(x)g(x) dx$ . Two functions are orthogonal if their inner product is zero.
Fourier Series (3.2.1): For a $2\pi$ -periodic function $f$ , the series is: $f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos(nx) + b_n \sin(nx))$ * Euler Formulas: * $a_0 = \frac{1}{\pi} \int_a^{a+2\pi} f(x) dx$ * $a_n = \frac{1}{\pi} \int_a^{a+2\pi} f(x) \cos(nx) dx$ * $b_n = \frac{1}{\pi} \int_a^{a+2\pi} f(x) \sin(nx) dx$
Arbitrary Period 2l: $f(x) \sim \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos\frac{n\pi x}{l} + b_n \sin\frac{n\pi x}{l})$
Special Cases: * Even Functions: Only cosine terms exist ( $b_n = 0$ ). $a_n = \frac{2}{l} \int_0^l f(x) \cos\frac{n\pi x}{l} dx$ . * Odd Functions: Only sine terms exist ( $a_n = 0$ ). $b_n = \frac{2}{l} \int_0^l f(x) \sin\frac{n\pi x}{l} dx$ .
Dirichlet Convergence Theorem (3.3.1): If $f$ and $f'$ are piecewise continuous and $f$ is periodic, the series converges to $f(x)$ at continuity points and to the average of limits $\frac{f(c^+) + f(c^-)}{2}$ at discontinuities.
Parseval's Identity (3.7): $\frac{1}{l} \int_{-l}^l [f(x)]^2 dx = \frac{a_0^2}{2} + \sum_{n=1}^{\infty} (a_n^2 + b_n^2)$ .
Fourier Transform: $f(w) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-iwt} dt$ .

Chapter 5: Limits and Continuity in Multiple Variables

Definition 5.1.1: A function of two variables assigns a unique number $f(x, y)$ to each pair $(x, y)$ in domain $D$ .
Limit Definition (5.3.2): $\lim_{(x,y) \to (a,b)} f(x, y) = L$ if \forall \varepsilon > 0, \exists \delta > 0 s.t. 0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta \implies |f(x, y) - L| < \varepsilon.
Path Dependency: If $f(x, y)$ approaches different values along different paths to $(a, b)$ , the limit does not exist.
Continuity (5.4.2): $f$ is continuous at $(a, b)$ if $\lim_{(x,y) \to (a,b)} f(x, y) = f(a, b)$ .
Squeeze Theorem (5.3.1): If $f \le g \le h$ and $\lim f = \lim h = L$ , then $\lim g = L$ .

Chapter 6: Partial Derivatives and Applications

Definition 6.1.1: * With respect to $x$ : $f_x(x, y) = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}$ . * With respect to $y$ : $f_y(x, y) = \lim_{h \to 0} \frac{f(x, y+h) - f(x, y)}{h}$ .
Clairaut's Theorem (6.2.1): If $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$ .
Chain Rule: * One independent variable: $\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$ .
Implicit Differentiation (6.5.1): If $f(x, y, z) = 0$ , then $\frac{\partial z}{\partial x} = -\frac{f_x}{f_z}$ and $\frac{\partial z}{\partial y} = -\frac{f_y}{f_z}$ .
Directional Derivative (6.6.1): In direction of unit vector $\mathbf{u} = (a, b)$ , $D_{\mathbf{u}}f = f_x a + f_y b = \nabla f \cdot \mathbf{u}$ .
Gradient Vector: $\nabla f = (f_x, f_y)$ . * $f$ increases most rapidly in the direction of $\nabla f$ . * Maximum rate of change is $||\nabla f||$ .
Tangent Plane (6.7.1): For $z = f(x, y)$ , the equation is $z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$ .
Optimization: * Second Partials Test: $D = f_{xx}f_{yy} - (f_{xy})^2$ . * D > 0 and f_{xx} > 0: Relative minimum. * D > 0 and f_{xx} < 0: Relative maximum. * D < 0: Saddle point. * $D = 0$ : Inconclusive. * Lagrange Multipliers: To maximize/minimize $f$ subject to $g = k$ , solve $\nabla f = \lambda \nabla g$ .

Chapter 7: Multiple Integrals

Fubini’s Theorem (7.1.1): Allows evaluation of double integrals as iterated integrals over rectangles: $\iint_R f(x, y) dA = \int_a^b \int_c^d f(x, y) dy dx = \int_c^d \int_a^b f(x, y) dx dy$ .
Polar Coordinates Integration: $\iint_R f(x, y) dA = \int_{\alpha}^{\beta} \int_a^b f(r \cos \theta, r \sin \theta) r dr d\theta$ .
Surface Area (7.1.7): $SA = \iint_R \sqrt{(f_x)^2 + (f_y)^2 + 1} dA$ .
Triple Integrals: * Cylindrical: Uses $(r, \theta, z)$ , volume element $dV = r dz dr d\theta$ . * Spherical: Uses $(\rho, \theta, \phi)$ , volume element $dV = \rho^2 \sin \phi d\rho d\theta d\phi$ .
Change of Variables (7.3.1): $\iint_R f(x, y) dx dy = \iint_S f(x(u, v), y(u, v)) |J(u, v)| du dv$ where $J(u, v) = \frac{\partial(x,y)}{\partial(u,v)}$ is the Jacobian.