Mathematical Tools for Physics

Infinite Series

Infinite series are among the most powerful and useful tools that you’ve been introduced to in an introductory
calculus course. It’s easy to get the impression that they are simply a clever exercise in manipulating limits and in
studying convergence, but they are among the majors tools used in analyzing differential equations, in developing
methods of numerical analysis, in defining new functions, in estimating the behavior of functions, and more.

2.1 The Basics
There are a handful of infinite series that you should have memorized and should know just as well as you do the
multiplication table. The first of these is the geometric series,

1 +x+x^2 +x^3 +x^4 +···=

∑∞

0

xn=

1

1 −x

for|x|< 1. (1)

It’s very easy derive because in this case it’s easy to sum the finite form of the series and then to take a limit.
Write the series out to the termxN and multiply it by(1−x).

(1 +x+x^2 +x^3 +···+xN)(1−x) =

(1 +x+x^2 +x^3 +···+xN)−(x+x^2 +x^3 +x^4 +···+xN+1) = 1−xN+1 (2)

If|x|< 1 then asN → ∞this last term,xN+1, goes to zero and you have the answer. Ifxis outside this
domain the terms of the infinite series don’t even go to zero, so there’s no chance for the series to converge to
anything.

The finite sum up toxN is useful on its own. For example it’s what you use to compute the payments on
a loan that’s been made at some specified interest rate. You use it to find the pattern of light from a diffraction
grating.

∑N

0

xn=

1 −xN+1

1 −x

30