Math Review Week 7

Highlights of what I (re)learned in week #7 of my math studies.

• Area bounded by two polar curves
• Infinite sequences and series

Area Bounded By Two Polar Curves

Let \(R\) be the region inside the polar curve \(r = 3\cos\theta\) and outside the polar curve \(r = 1+\cos\theta\).

Finding the Intersection Angles

To find where the curves intersect, set their equations equal:

\[ 3\cos\theta=1+\cos\theta. \]

Solving for \(\theta\),

\[ 2\cos\theta=1 \]

and therefore

\[ \cos\theta=\frac12. \]

The intersection angles in the interval \([-\pi,\pi]\) are

\[ \theta=-\frac{\pi}{3} \qquad\text{and}\qquad \theta=\frac{\pi}{3}. \]

The corresponding intersection points in Cartesian coordinates are

\[ \left(\frac34,\frac{3\sqrt3}{4}\right) \qquad\text{and}\qquad \left(\frac34,-\frac{3\sqrt3}{4}\right). \]

Compute the area via integration

\[ A=\frac12\int_{-\pi/3}^{\pi/3} \left[9\cos^2\theta-(1+\cos\theta)^2\right]\,d\theta \]

Infinite sequences and series

Geometric Series

\[ \sum_{k=0}^{\infty} a\cdot r^k, \text{when } |r| < 1, \text{ converges to } \frac{a}{1-r} \]

Divergence Test

\[ \text{If the }\lim_{n \to \infty} a_n \ne 0 \text{, then} \sum_{n=1}^{\infty} \text{will diverge} \]

The Integral Test

If \(f(x)\) positive, continuous and decreasing on \([k, \infty)\):

(1) if \(\int_{k}^{\infty} f(x)\; dx\) is convergent, then \(\sum_{n=k}^{\infty} f(n)\) is also convergent.

(2) if \(\int_{k}^{\infty} f(x)\; dx\) is divergent, then \(\sum_{n=k}^{\infty} f(n)\) is also divergent.

Harmonic Series (special case of P-Series)

Harmonic Series = \(\sum_{n=1}^{\infty} \frac{1}{n} =\) \(1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots\) diverges

P-Series

\[ \sum_{n=1}^{\infty} \frac{1}{n^p} \text{ if } p > 1 \text{, converges; if } p \le 1 \text{, diverges} \]

Direct Comparison Test

\[ \text{Let A }= \sum_{n=1}^{\infty} a_n \text{ , and B } = \sum_{n=1}^{\infty} b_n \text{ ,where } a_n, b_n \ge 0; \; a_n \le b_n \text{ , for } n \in [1, \infty) \]

(1) If B (larger) converges, then A (smaller) also converges.

(2) If A (smaller) diverges, then B (larger) also diverges.

Limit Comparison Test

\[ \text{Consider } \sum_{n=k}^{\infty} a_n \text{ and } \sum_{n=k}^{\infty} b_n \text{ where } a_n \ge 0 \text{, } b_n > 0 \text{ for all } n=k, k+1, \cdots \]

\[ \text{If } \lim_{n \to \infty} \frac{a_n}{b_n} \text{ is positive and finite, then either both converge, or both diverge} \]

Alternating Series Test

\[ \sum_{n=k}^{\infty} a_n \text{ If } a_n \text{ can be written as either } (-1)^n\cdot b_n or (-1)^{n+1}\cdot b_n \]

\[ \text{where } b_n \ge 0 \text{, } \lim_{n \to \infty} b_n = 0 \text{, } \{ b_n \} \text{ decreasing sequence} \]

\[ \text{Then } \sum_{n=k}^{\infty} a_n \text{ converges} \]

Ratio Test

\[ \sum_{n=k}^{\infty} a_n \text{ if we take } \lim_{n \to \infty} | \frac{a_{n+1}}{a_n}| = L \]

If \(L < 1\), the series converges

If \(L > 1\), the series diverges

If \(L = 1\), the test is inconclusive

Conditional and Absolute Convergence

If a series converges, but its absolute value does not converge, we say the series converges conditionally.

If a series converges, and its absolute value also converges, we say it converges absolutely.