Calculus II: Series, Parametric Equations, and Polar Coordinates Study Guide

Sequences and Series

Sec. 9.1 Sequences

A sequence is an ordered list of numbers, typically defined by a formula for its nth term. Sequences are foundational in calculus for understanding limits and convergence.

• Definition: A sequence is a function whose domain is the set of positive integers.

• Limit of a Sequence: The value that the terms of a sequence approach as n becomes very large.

• Example: The sequence $a_n = \frac{1}{n}$ converges to 0 as $n \to \infty$.

Sec. 9.2 Infinite Series

An infinite series is the sum of the terms of a sequence. The study of series involves determining whether the sum converges to a finite value.

• Definition: $\sum_{n=1}^{\infty} a_n$

• Convergence: A series converges if the sequence of its partial sums approaches a finite limit.

• Example: The geometric series $\sum_{n=0}^{\infty} ar^n$ converges if $|r| < 1$.

Sec. 9.3 The Integral Test

The Integral Test is a method for determining the convergence of series whose terms are positive and decrease monotonically.

• Statement: If $f(n) = a_n$ is positive, continuous, and decreasing for $n \geq 1$, then $\sum_{n=1}^{\infty} a_n$ converges if and only if $\int_1^{\infty} f(x) dx$ converges.

• Example: $\sum_{n=1}^{\infty} \frac{1}{n^p}$ converges for $p > 1$.

Sec. 9.4 The Direct Comparison Test

The Direct Comparison Test compares a given series to a known benchmark series to determine convergence or divergence.

• Statement: If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, then $\sum a_n$ converges.

• Example: Compare $\sum \frac{1}{n^2+1}$ to $\sum \frac{1}{n^2}$.

Sec. 9.5 Absolute Convergence; The Ratio and Root Tests

Absolute convergence means a series converges even when all terms are replaced by their absolute values. The Ratio Test and Root Test are standard tools for testing convergence.

• Ratio Test: For $\sum a_n$, compute $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$. If $L < 1$, the series converges absolutely.

• Root Test: For $\sum a_n$, compute $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. If $L < 1$, the series converges absolutely.

• Example: $\sum \frac{n!}{n^n}$

Sec. 9.6 Alternating Series, Absolute and Conditional Convergence

An alternating series has terms that alternate in sign. Conditional convergence occurs when a series converges, but not absolutely.

• Alternating Series Test: If $a_n$ decreases to 0, $\sum (-1)^n a_n$ converges.

• Absolute vs. Conditional: $\sum a_n$ is absolutely convergent if $\sum |a_n|$ converges; conditionally convergent if $\sum a_n$ converges but $\sum |a_n|$ diverges.

• Example: $\sum (-1)^{n+1} \frac{1}{n}$ converges conditionally.

Sec. 9.7 Power Series

A power series is an infinite series in the form $\sum_{n=0}^{\infty} c_n(x-a)^n$. Power series are used to represent functions.

• Radius of Convergence: The interval around $a$ where the series converges.

• Example: $\sum_{n=0}^{\infty} x^n$ converges for $|x| < 1$.

Sec. 9.8 Taylor and Maclaurin Series

Taylor series represent functions as infinite sums of derivatives at a point. Maclaurin series are Taylor series centered at $a=0$.

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

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

• Example: $e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}$

Parametric Equations and Polar Coordinates

Sec. 10.1 Parametrizations of Plane Curves

Parametric equations describe curves by expressing coordinates as functions of a parameter, usually $t$.

• Definition: $x = f(t), y = g(t)$

• Example: The circle: $x = r \cos t, y = r \sin t$ for $0 \leq t < 2\pi$

Sec. 10.2 Calculus with Parametric Curves

Calculus can be applied to parametric curves to find slopes, areas, and arc lengths.

• Derivative: $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$

• Arc Length: $L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt$

Sec. 10.3 Polar Coordinates

Polar coordinates represent points in the plane using a radius and angle $(r, \theta)$.

• Conversion: $x = r \cos \theta$, $y = r \sin \theta$

• Example: The circle $r = a$

Sec. 10.5 Areas and Lengths in Polar Coordinates

Calculus in polar coordinates allows computation of areas and arc lengths for curves defined by $r = f(\theta)$.

• Area: $A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta$

• Arc Length: $L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} d\theta$

Additional info: These notes are structured from section headings, typical of a Calculus II syllabus or textbook, and expanded with standard academic content for each topic.