Comprehensive Multivariable Calculus and Infinite Series Study Guide

Sequences of Real Numbers

• Definition: A sequence of real numbers is a function $f: \nu \rightarrow \mathbb{R}$ where $f(n)=a_{n}ppp$ , represented as $\{a_n\}$ .

• Convergence: A sequence $\{a_n\}$ converges to $L$ ($\lim_{n \rightarrow \infty} a_n = L$) if for every $\epsilon > 0$, there exists an integer $N$ such that $\|a_n - L\| < \epsilon$ for all $n > N$. Otherwise, it diverges.

• Boundedness:

  • Bounded from Above / Below: Bounded above if $a_n \le M$; bounded below if $m \le a_n$.

  • Supremum (LUB) & Infimum (GLB): Least upper bound $M$ and greatest lower bound $m$.

  • Bounded Sequence: Bounded both from above and below ($m \le a_n \le M$).

• Monotonicity: Non-decreasing if $a_n \le a_{n+1}$; non-increasing if $a_n \ge a_{n+1}$.

• Key Theorems:

  • Monotone Convergence Theorem: A monotonic sequence converges if and only if it is bounded.

  • Algebraic Limit Laws: Limits preserve sums, differences, constant multiples, products, and quotients (provided the denominator limit is non-zero).

  • Continuous Function Theorem: If $a_n \rightarrow L$ and $f(x)$ is continuous at $L$, then $f(a_n) \rightarrow f(L)$.

  • Sandwich (Squeeze) Theorem: If $a_n \le b_n \le c_n$ and $\lim_{n \rightarrow \infty} a_n = \lim_{n \rightarrow \infty} c_n = L$, then $\lim_{n \rightarrow \infty} b_n = L$.

  • Subsequence Theorem: If even $\{a_{2n}\}$ and odd $\{a_{2n+1}\}$ subsequences converge to $L$, then $\{a_n\} \rightarrow L$.

  • Boundedness vs. Convergence: Every convergent sequence is bounded, but bounded sequences are not necessarily convergent.

Infinite Series

• Definitions: An infinite series $\sum_{n=1}^{\infty} a_n$ has partial sums $S_n = \sum_{k=1}^n a_k$. It converges to sum $L$ if $\lim_{n \rightarrow \infty} S_n = L$.

• Geometric Series: $\sum_{n=1}^{\infty} a r^{n-1}$ converges to $\frac{a}{1 - r}$ if $\|r\| < 1$; diverges if $\|r\| \ge 1$.

• $n$-th Term Test for Divergence: If $\lim_{n \rightarrow \infty} a_n \neq 0$ (or does not exist), then $\sum a_n$ diverges. If the limit is $0$, the test is inconclusive.

Tests for Convergence of Series

• Integral Test: For a positive, continuous, decreasing function $f(x)$ with $f(n) = a_n$, the series $\sum_{n=N}^{\infty} a_n$ and improper integral $\int_N^{\infty} f(x)\,dx$ both converge or both diverge.

  • $p$-Series: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \le 1$.

• Direct Comparison Test (DCT): If $0 \le a_n \le c_n$ and $\sum c_n$ converges, $\sum a_n$ converges. If $0 \le d_n \le a_n$ and $\sum d_n$ diverges, $\sum a_n$ diverges.

• Limit Comparison Test (LCT): For $a_n, b_n > 0$, if $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = c \in (0, \infty)$, both series behave identically.

• Ratio Test: Let $\rho = \lim_{n \rightarrow \infty} \left|\frac{a_{n+1}}{a_n}\right|$. Converges if $\rho < 1$, diverges if $\rho > 1$, inconclusive if \rho = 1$.\n- **Root Test:** Let \rho = \lim_{n \rightarrow \infty} \sqrt[n]{|a_n|}$. Converges if$\rho < 1$, diverges if$\rho > 1$, inconclusive if$\rho = 1$.

Alternating Series, Absolute, and Conditional Convergence

• Leibniz Theorem (Alternating Series Test): $\sum_{n=1}^{\infty} (-1)^{n+1} u_n$ ($u_n > 0$) converges if $u_n$ is non-increasing ($u_n \ge u_{n+1}$) and \lim_{n \rightarrow \infty} u_n = 0$.\n- **Convergence Types:**\n - **Absolute Convergence:** \sum |a_n|$converges (implies$\sum a_n converges).\n - **Conditional Convergence:** \sum a_n$converges, but$\sum |a_n| diverges.\n\n### Power Series, Taylor, and Maclaurin Series\n\n- **Power Series:** \sum_{n=0}^{\infty} c_n (x - a)^n$centered at$a$. Converges only at$x = a$($R = 0$), on an interval$|x - a| < R$, or for all$x$($R = \infty).\n- **Radius of Convergence (R$):** Calculated via$R = \lim_{n \rightarrow \infty} \left|\frac{c_n}{c_{n+1}}\right|$or$R = \frac{1}{\lim_{n \rightarrow \infty} \sqrt[n]{|c_n|}}$.

• Taylor & Maclaurin Series:

  • Taylor Series at $x = a$: $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n$

  • Maclaurin Series at $x = 0$: $\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$

Vector-Valued Functions and Motion in Space

• Vector Function: \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle$.\n- **Kinematics:**\n - **Velocity:** \mathbf{v}(t) = \mathbf{r}'(t)\n - **Speed:** |\mathbf{v}(t)| = \frac{ds}{dt}\n - **Acceleration:** \mathbf{a}(t) = \mathbf{v}'(t) = \mathbf{r}''(t)\n - **Direction of Motion:** \frac{\mathbf{v}(t)}{|\mathbf{v}(t)|}\n- **Constant Length Property:** If |\mathbf{r}(t)| = c$, then$\mathbf{r}(t) \cdot \mathbf{r}'(t) = 0 (orthogonal).\n\n### Arc Length, Unit Tangent, Curvature, and Normal Vector\n\n- **Arc Length:** L = \int_a^b |\mathbf{v}(t)|\,dt\n- **Unit Tangent Vector:** \hat{\mathbf{T}}(t) = \frac{\mathbf{v}(t)}{|\mathbf{v}(t)|}$

• Curvature: $\kappa = \left\|\frac{d\hat{\mathbf{T}}}{ds}\right\| = \frac{1}{\|\mathbf{v}(t)\|}\left\|\frac{d\hat{\mathbf{T}}}{dt}\right\|$

  • Explicit Plane Curve $y = f(x)$: $\kappa = \frac{\|f''(x)\|}{\left(1 + (f'(x))^2\right)^{3/2}}$

  • Circle of Radius $a$: $\kappa = \frac{1}{a}$; Straight Line: $\kappa = 0$

• Principal Unit Normal Vector: $\hat{\mathbf{N}} = \frac{1}{\kappa}\frac{d\hat{\mathbf{T}}}{ds} = \frac{d\hat{\mathbf{T}}/dt}{\|d\hat{\mathbf{T}}/dt\|}$, pointing toward the concave side.

• Radius of Curvature: $\rho = \frac{1}{\kappa}$

Polar Coordinates and Polar Graphing

• Transformations: $x = r\cos(\theta)$, $y = r\sin(\theta)$, $r^2 = x^2 + y^2$, $\tan(\theta) = \frac{y}{x}$.

• Non-Uniqueness: $(r, \theta) = (r, \theta + 2n\pi) = (-r, \theta + (2n + 1)\pi)$.

• Slope of Polar Curve: $\frac{dy}{dx} = \frac{f'(\theta)\sin(\theta) + f(\theta)\cos(\theta)}{f'(\theta)\cos(\theta) - f(\theta)\sin(\theta)}$. At the pole ($r = 0$), the slope is $\tan(\theta_0)$.

• Symmetry:

  • $x$-axis: Replace $\theta$ with $-\theta$.

  • $y$-axis: Replace $\theta$ with $\pi - \theta$.

  • Origin: Replace $r$ with $-r$ or $\theta$ with $\theta + \pi$.