Mathematical Tools for Physics

2—Infinite Series 33

This shows the terms of the series for the sine as stated.
Does this show that the series converges? If it converges does it show that it converges to the sine? No
to both. Each statement requires more work, and I’ll leave the second one to advanced calculus books. Even
better, when you understand the subject of complex variables, these questions about series become much easier
to understand.
The generalization to any function is obvious. You match the coefficients in the assumed expansion, and
get

f(x) =f(0) +xf′(0) +

x^2

2!

f′′(0) +

x^3

3!

f′′′(0) +

x^4

4!

f′′′′(0) +···

You don’t have to do the expansion about the pointx= 0. Do it aboutx 0 instead.

f(x) =f(x 0 ) + (x−x 0 )f′(x 0 ) +

(x−x 0 )^2

2!

f′′(x 0 ) +··· (4)

What good are infinite series?
This is sometimes the way that a new function is introduced and developed, typically by determining a series
solution to a new differential equation.
This is a tool for the numerical evaluation of functions.
This is an essential tool to understand and invent numerical algorithms for integration, differentiation, interpola-
tion, and many other common numerical methods.
To understand the behavior of complex-valued functions of a complex variable you will need to understand these
series for the case that the variable is a complex number.
All the series that I’ve written above are power series (Taylor series), but there are many other possibilities.

ζ(z) =

∑∞

1

1

nz

(5)

x^2 =

L^2

3

+

2 L^2

π^2

∑∞

1

(−1)n

1

n^2

cos

(nπx

L

)

(−L≤x≤L) (6)

The first is a Dirichlet series defining the Riemann zeta function, a function that appears in statistical mechanics
among other places.