Advanced Placement Calculus BC including series, parametric equations, and polar functions.
A power series expresses a function as an infinite sum of powers of \(x\):
\[ f(x) = a_0 + a_1(x-c) + a_2(x-c)^2 + \cdots \]
The most famous power series are Taylor and Maclaurin series, which help us approximate functions like sine, cosine, and exponential.
The Taylor series for a function \(f(x)\) centered at \(c\):
\[ f(x) = f(c) + f'(c)(x-c) + \frac{f''(c)}{2!}(x-c)^2 + \cdots \]
If \(c = 0\), it's called a Maclaurin series.
Power series can approximate complicated functions and are used in everything from calculators to computer graphics and scientific modeling.
The Maclaurin series for \(e^x\) is \(1 + x + x^2/2! + x^3/3! + \cdots\).
The sine function can be written as \(x - x^3/3! + x^5/5! - \cdots\).
Power and Taylor series turn complex functions into infinite sums for easier calculation and estimation.