AP Calc BC Part 2 Notes

Arithmetic Sequence Term

$t_n = a + (n-1)d$ (where $a$ is the first term and $d$ is the common difference)

Sum of an Arithmetic Series

$$S_n = \frac{n}{2}(2a + (n-1)d)$$

Geometric Sequence Term

$t_n = a r^{n-1}$ (where $a$ is the first term and $r$ is the common ratio)

Sum of a Finite Geometric Series

$$S_n = a\frac{1 - r^n}{1 - r}$$

Sum of an Infinite Geometric Series

$S = \frac{a}{1-r}$, provided that $\|r\| < 1$

Absolute Convergence

Both the original series $\sum a_n$ and the absolute value series $\sum \|a_n\|$ converge.

Conditional Convergence

The original series $\sum a_n$ converges, but the absolute value series $\sum \|a_n\|$ diverges.

Integral Test Conditions

If $f(x)$ is positive, continuous, and decreasing for $x \ge 1$ where $f(k) = a_k$, then $\int_{1}^{\infty} f(x) \, dx$ and $\sum_{k=1}^{\infty} a_k$ either both converge or both diverge.

P-Series Test

$\sum_{k=1}^{\infty} \frac{1}{k^p}$ converges if $p > 1$; diverges if $0 < p \le 1$.

Direct Comparison Test

If $0 \le a_n \le b_n$, then if $\sum b_k$ converges, so does $\sum a_k$. If $\sum a_k$ diverges, then $\sum b_k$ diverges as well.

Ratio Test

Evaluate $L = \lim_{k \to \infty} \left\| \frac{a_{k+1}}{a_k} \right\|$. If $L < 1$, the series converges absolutely; if $L > 1$, it diverges; if $L = 1$, the test is inconclusive.

n-th Term Test for Divergence

If $\lim_{k \to \infty} a_k \neq 0$, then the series $\sum a_k$ diverges.

Alternating Series Test Conditions

1. $a_k > 0$ for all $k$; 2. $\lim_{k \to \infty} a_k = 0$; 3. $a_{k+1} \le a_k$ (the terms are non-increasing).

Geometric Series Test

$\sum a r^n$ converges if $\|r\| < 1$; diverges if $\|r\| \ge 1$.

Limit Comparison Test

Evaluate $c = \lim_{k \to \infty} \frac{a_k}{b_k}$. If $0 < c < \infty$, then both series either both converge or diverge.

General Maclaurin Series

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

Maclaurin Series for sin(x)

$$\sin(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)!} x^{2k+1}$$

Maclaurin Series for cos(x)

$$\cos(x) = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k)!} x^{2k}$$

Maclaurin Series for e^x

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

Maclaurin Series for \frac{1}{1-x}

$\frac{1}{1-x} = \sum_{n=0}^{\infty} x^n$ valid for $\|x\| < 1$.

General Taylor Series

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

Alternating Series Error Bound

$\|S - S_k\| \le a_{k+1}$.

Lagrange Error Bound

$\|E_n(x)\| \le \frac{M}{(n+1)!} \|x-a\|^{n+1}$.

Interval of Convergence Three Possibilities

1. Converges only at $x = a$ ($R = 0$); 2. Converges absolutely for all $x$ ($R = \infty$); 3. Converges for $\|x-a\| < R$ and diverges for $\|x-a\| > R$, yielding interval $(a-R, a+R)$.

Steps to Find Interval of Convergence

1. Apply the Ratio Test to solve for $\|x-a\| < R$; 2. Test each endpoint $x = a-R$ and $x = a+R$ for convergence.

First Derivative for Parametric Equations

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$.

Second Derivative for Parametric Equations

$\frac{d^2y}{dx^2} = \frac{\frac{d}{dt}\left(\frac{dy}{dx}\right)}{\frac{dx}{dt}}$.

Rectangular Coordinate Equations for Polar Coordinates

$x = r \cos(\theta)$, $y = r \sin(\theta)$, and $x^2 + y^2 = r^2$.

Polar Curve Derivative

$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{\frac{dr}{d\theta}\sin(\theta) + r\cos(\theta)}{\frac{dr}{d\theta}\cos(\theta) - r\sin(\theta)}$.

Area Bounded by a Polar Curve

$A = \frac{1}{2} \int_{a}^{b} r^2 \, d\theta$.

Area Between Two Polar Curves

$A = \frac{1}{2} \int_{a}^{b} \left((r_2)^2 - (r_1)^2\right) d\theta$.

Cartesian Arc Length

$L = \int_{a}^{b} \sqrt{1 + [f'(x)]^2} \, dx$ (for $y = f(x)$).

Cartesian Arc Length for x = g(y)

$L = \int_{c}^{d} \sqrt{1 + [g'(y)]^2} \, dy$.

Parametric Arc Length

$L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$.