Power Series: Taylor & Maclaurin Series

Maclaurin series: special case of Taylor series centered at $x=0$ $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$$
Taylor series: expansion of a function about a point $a$ $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$$

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots$
- For $\ln(1-x)$: substitute $x \to -x$ $\Rightarrow$ $\ln(1-x) = -x - \frac{x^2}{2} - \frac{x^3}{3} - \frac{x^4}{4} - \cdots$
- For $\ln(2+x)$: write $\ln[2(1+\frac{x}{2})] = \ln 2 + \ln(1+\frac{x}{2})$ and expand
$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$
$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

Proving identities: Expand both sides as series (e.g., $\cos 2x = 1 - 2\sin^2 x$)
Definite integrals: Approximate $\int_0^{0.5} \frac{1}{\sqrt{1+x^2}} dx$ by truncating Maclaurin series
Series multiplication: Find a specific coefficient by multiplying expansions

Circle Geometry

General: $x^2 + y^2 + Dx + Ey + F = 0$
Complete the square to get standard form: $(x-h)^2 + (y-k)^2 = r^2$
- Centre: $(h,k)$
- Radius: $r = \sqrt{h^2 + k^2 - F}$

Distance between centre $C$ and point $P$: $d = \sqrt{(x_P - x_C)^2 + (y_P - y_C)^2}$
Circle through three points: Solve system from $x^2+y^2+Dx+Ey+F=0$

Line $y = mx + c$ is tangent to circle if perpendicular distance from centre to line equals radius
- Distance from $(h,k)$ to $Ax+By+C=0$: $\frac{|Ah+Bk+C|}{\sqrt{A^2+B^2}}$

For circles with centres $C_1, C_2$ and radii $r_1, r_2$, let $d = |C_1 C_2|$: