Advanced Series Applications Capstone — Theory
1. Why series are useful in applications
Series allow complicated functions to be approximated by simpler finite sums. This is useful in physics,
engineering, differential equations, and signal processing.
\[
f(x)\approx P_K(x)=\sum_{n=0}^{K}T_n(x).
\]
2. Error of an approximation
The approximation error is the difference between the exact model and the finite series:
\[
E_K(x)=f(x)-P_K(x).
\]
At a selected point \(x_0\), this becomes
\[
E_K(x_0)=f(x_0)-P_K(x_0).
\]
3. Differential equations: exponential decay
A basic first-order differential equation is
\[
y'=-ky.
\]
With \(y(0)=1\), the solution is
\[
y=e^{-kx}.
\]
Its Taylor series is
\[
e^{-kx}
=
\sum_{n=0}^{\infty}\frac{(-kx)^n}{n!}.
\]
4. Physics: harmonic oscillator
The simple harmonic oscillator is connected to the differential equation
\[
y''+\omega^2y=0.
\]
One solution is
\[
y=\cos(\omega x).
\]
The series approximation is
\[
\cos(\omega x)
=
\sum_{n=0}^{\infty}
(-1)^n\frac{(\omega x)^{2n}}{(2n)!}.
\]
5. Physics: small-angle pendulum
The pendulum equation contains \(\sin\theta\). For small angles, this can be approximated by a series:
\[
\sin\theta
=
\theta-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots.
\]
The simplest small-angle approximation is
\[
\sin\theta\approx \theta.
\]
Higher-order terms improve the approximation for larger angles.
6. Signal processing: square wave Fourier series
A square wave can be approximated by adding sine waves:
\[
f(x)
\sim
\frac{4A}{\pi}
\sum_{n=0}^{\infty}
\frac{\sin((2n+1)x)}{2n+1}.
\]
Each term is an odd harmonic. Adding more harmonics makes the wave look more square,
except near jump discontinuities where overshoot may occur.
7. Physics: relativistic gamma approximation
The relativistic factor is
\[
\gamma=\frac{1}{\sqrt{1-x^2}},
\qquad x=\frac{v}{c}.
\]
Its power series is
\[
\gamma
=
1+\frac12x^2+\frac38x^4+\frac5{16}x^6+\cdots,
\qquad |x|<1.
\]
8. How the calculator builds \(P_K(x)\)
For each application, the calculator creates terms \(T_n(x)\), then adds them:
\[
P_K(x)=T_0(x)+T_1(x)+T_2(x)+\cdots+T_K(x).
\]
At \(x_0\), the numeric approximation is
\[
P_K(x_0)=\sum_{n=0}^{K}T_n(x_0).
\]
9. Connection to Multivariable Calculus
Series become even more powerful in Multivariable Calculus. A single-variable Taylor polynomial becomes
a multivariable Taylor approximation:
\[
f(x,y)\approx f(a,b)
+f_x(a,b)(x-a)+f_y(a,b)(y-b)+\cdots.
\]
This idea is used for local linearization, optimization, and physical field approximations.
10. Connection to Differential Equations
Some differential equations cannot be solved easily by elementary formulas.
Power series methods assume a solution of the form
\[
y=\sum_{n=0}^{\infty}a_nx^n.
\]
Substituting this into the differential equation produces equations for the coefficients \(a_n\).
12. Common mistakes
- Using the approximation too far from the expansion center: Taylor polynomials often work best near the center.
- Ignoring convergence limits: some power series only converge on a finite interval.
- Confusing exact and approximate models: \(P_K(x)\) is only a finite approximation.
- Using too few terms in engineering models: low-order approximations may miss important behavior.
- Expecting Fourier series to match jumps exactly: at discontinuities, Fourier series converge to midpoint values.
