Advanced Integration, Sequences, Series, and Parametric Equations

Improper Integrals

Definition and Evaluation

Improper integrals are definite integrals where either the interval of integration is infinite or the integrand becomes infinite within the interval. These integrals are evaluated as limits.

• Type 1: Infinite limits of integration (e.g., $\int_a^{\infty} f(x)\,dx$)

• Type 2: Integrand has an infinite discontinuity (e.g., $\int_a^b \frac{1}{(x-c)^p}\,dx$ where $c$ is in $[a, b]$)

General Approach:

• Replace the problematic bound with a variable, take the limit as the variable approaches the bound.

Example:

Evaluate $\int_1^{\infty} \frac{1}{x^2}\,dx$

$\int_1^{\infty} \frac{1}{x^2}\,dx = \lim_{b \to \infty} \int_1^b \frac{1}{x^2}\,dx = \lim_{b \to \infty} [-\frac{1}{x}]_1^b = \lim_{b \to \infty} ( -\frac{1}{b} + 1 ) = 1$

Comparison Test for Improper Integrals

The comparison test helps determine the convergence or divergence of improper integrals by comparing with a known function.

• If $0 \leq f(x) \leq g(x)$ for all $x$ in $[a, \infty)$ and $\int_a^{\infty} g(x)\,dx$ converges, then $\int_a^{\infty} f(x)\,dx$ also converges.

• If $\int_a^{\infty} f(x)\,dx$ diverges and $f(x) \geq g(x) \geq 0$, then $\int_a^{\infty} g(x)\,dx$ also diverges.

Example: Compare $\int_1^{\infty} \frac{1}{x^2+1}\,dx$ with $\int_1^{\infty} \frac{1}{x^2}\,dx$.

Separable Differential Equations

Definition and Solution Method

A separable differential equation can be written as $\frac{dy}{dx} = f(x)g(y)$, allowing variables to be separated on each side.

• Rewrite as $\frac{1}{g(y)}dy = f(x)dx$

• Integrate both sides: $\int \frac{1}{g(y)}dy = \int f(x)dx$

• Solve for $y$ if possible.

Example: Solve $\frac{dy}{dx} = xy$

$\frac{1}{y}dy = xdx$ $\int \frac{1}{y}dy = \int xdx$ $\ln|y| = \frac{1}{2}x^2 + C$ $y = Ce^{\frac{1}{2}x^2}$

Sequences

Definition and Convergence

A sequence is an ordered list of numbers, often defined by a formula $a_n$. A sequence converges if it approaches a finite limit as $n \to \infty$.

• Limit of a Sequence: $\lim_{n \to \infty} a_n = L$

• If the limit exists and is finite, the sequence converges; otherwise, it diverges.

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

Infinite Series

Definition

An infinite series is the sum of the terms of a sequence: $\sum_{n=1}^{\infty} a_n$.

Integral Test

If $a_n = f(n)$ where $f$ is positive, continuous, and decreasing for $n \geq N$, then:

• If $\int_N^{\infty} f(x)dx$ converges, so does $\sum_{n=N}^{\infty} a_n$.

• If the integral diverges, so does the series.

Comparison Tests

Used to compare a series with another whose convergence is known.

• Direct Comparison Test: Compare $a_n$ with $b_n$ term by term.

• Limit Comparison Test: If $\lim_{n \to \infty} \frac{a_n}{b_n} = c$ (where $0 < c < \infty$), both series converge or diverge together.

Alternating Series, Absolute and Conditional Convergence

• Alternating Series: Terms alternate in sign, e.g., $\sum (-1)^{n} a_n$.

• Alternating Series Test: If $a_n$ decreases to 0, the series converges.

• Absolute Convergence: $\sum |a_n|$ converges.

• Conditional Convergence: $\sum a_n$ converges, but $\sum |a_n|$ diverges.

Ratio and Root Tests

• Ratio Test: Compute $L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

  • If $L < 1$, the series converges absolutely.

  • If $L > 1$ or $L = \infty$, the series diverges.

  • If $L = 1$, the test is inconclusive.

• Root Test: Compute $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$

  • Same conclusions as the Ratio Test.

Power Series and Taylor Series

Power Series

A power series is an infinite series of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$.

• Radius of Convergence: The set of $x$ values for which the series converges.

Taylor and Maclaurin Series

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

• Maclaurin Series: Special case of Taylor series at $a=0$.

Example: Maclaurin series for $e^x$:

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

Convergence of Taylor Series

• Determined by the radius of convergence.

• May not converge to $f(x)$ outside the interval of convergence.

Applications of Taylor Series

• Approximating functions

• Solving differential equations

• Evaluating limits

Parametric Equations and Polar Coordinates

Parametric Equations

Parametric equations express a curve using a parameter $t$:

• $x = f(t)$, $y = g(t)$

• Eliminate $t$ to find the Cartesian equation if needed.

Example: $x = \cos t$, $y = \sin t$ traces the unit circle.

Polar Coordinates

Points are represented as $(r, \theta)$, where $r$ is the distance from the origin and $\theta$ is the angle from the positive $x$-axis.

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

Graphing in Polar Coordinates

• Plot points for various $\theta$ values to sketch the curve.

• Common curves: circles, spirals, roses, lemniscates.

Areas and Lengths in Polar Coordinates

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

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

Conics in Polar Coordinates

Conic sections (ellipse, parabola, hyperbola) can be represented in polar form:

• General form: $r = \frac{ed}{1 + e\cos\theta}$, where $e$ is the eccentricity and $d$ is the directrix distance.