Applications of Series

Applying series simply means swapping a hard function for its Taylor polynomial so you can do the math with easy pieces. All we're really saying is that polynomials are the easiest things in calculus: you can integrate them, divide them and plug numbers into them without any tricks. The formula $\int_0^{b} e^{-x^2}\,dx = \sum \frac{(-1)^n\, b^{2n+1}}{n!\,(2n+1)}$ looks dense, but it's just the series for $e^{-x^2}$ (from the last lesson) with each power $x^{2n}$ integrated to $\frac{b^{2n+1}}{2n+1}$. Every step here is something you've already done, so bear with it.

Integrals. Back in the probability lesson we hit a wall: the bell curve $e^{-x^2}$ has no elementary antiderivative, so we had to rely on tables. Series get us past it. Expand the integrand, integrate term by term, and stop when the alternating series bound from the last unit says you're accurate enough. This is how those tables get made.

Limits. Near $x = 0$, replace each function by the start of its series:

One line, where L'Hôpital would take three rounds. Think about why it's so clean: the series shows you exactly which power of $x$ survives the subtraction, and that's the whole story of how fast the top goes to 0.

Products and quotients. Multiply series like long polynomials, keeping only the terms you need:

Approximations in science. $\sin\theta \approx \theta$ and $\cos\theta \approx 1 - \frac{\theta^2}{2}$ for small angles, and $e^x \approx 1 + x$ for small $x$. These are just the first terms of the series, and each one comes with an error bound from the next term, so you know how small "small" has to be.

The normal distribution's probabilities, the period of a real (large-swing) pendulum and the field of a magnetic dipole are all computed from series, because the exact integrals have no closed form.