Mathematical Tools for Physics - Department of Physics - University

Infinite Series

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Infinite series are among the most powerful and useful tools that you’ve encountered in your 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 memorize 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. (2.1)

It’s very easy derive because in this case you can sum the finite form of the series and then 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 (2.2)

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

(2.3)

Some other common series that you need to know are power series for elementary functions:

ex= 1 +x+

x^2

2!

+··· =

∑∞

0

xk

k!

sinx=x−

x^3

3!

+··· =

∑∞

0

(−1)k

x^2 k+1

(2k+ 1)!

cosx= 1−

x^2

2!

+··· =

∑∞

0

(−1)k

x^2 k

(2k)!

ln(1 +x) =x−

x^2

2

+

x^3

3

−··· =

∑∞

1

(−1)k+1

xk

k

(|x|<1) (2.4)

(1 +x)α= 1 +αx+

α(α−1)x^2

2!

+··· =

∑∞

k=0

α(α−1)···(α−k+ 1)

k!

xk (|x|<1)

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