Ch. 1- Infinite Series>Background
Chapter 1: Infinite Series
I. Background
• An infinite series is an expression of the form:

a
n
 a1  a 2  a 3  ...  a n  ...
n 1
where there is some rule for how the a’s are related to each other.
ex: 1) 1  2  3  4  ...
2)
1
2

1

4
1
 ...
8
3) 1  4  9  ...
4) x - x 
2
x
3
2
 ...
Ch. 1- Infinite Series>Background
• Why do physicists care about infinite series?
1) Loads of physics problems involve infinite series.
ex: Dropped ball- how far does it travel?
d h
h

2
h
4

h
 ...
8
ex: Swinging pendulum- how long until it stops swinging? (or will it ever stop?)
Tstop  4 s 
*picture?*
4s
3

4s
6
 ...
Ch. 1- Infinite Series>Convergence and Divergence
2) Complicated math expressions can be approximated by series and
then solved more easily.
ex:
e
x
cos( x ) dx 
 (1  x 
x
3

3
x
4
 ...) dx
6
II. Convergence and Divergence
• How do we know if a series has a finite sum? (eg. will the pendulum ever stop?)
defn: Mathematics terminology- The series converges if it has a finite sum;
otherwise, the series diverges.
defn: We define the sum of a series (if it has one) to be: S  lim S n
x 
n
where S n 
a
k
k 1
• Do all series for which

ex:
1
n
n 1
 1
1
2

1
3

1
4
is the sum of the first n terms of the series.
a n 1  a n
 ...
for all n converge? No!
doesn’t converge. It approaches zero too slowly.
(Proof in hw)
Ch. 1- Infinite Series>Convergence and Divergence>Geometric Series
A. Geometric series
•
Each term is multiplied by a fixed number to get the next term.
a  ar  ar  ar  ...
2
3
ex: 1) 1 + 3 + 9 + 27 + ...
2) 2 – 5 + 18 – 54 + ...
•
We can show that only for a geometric series, the sum of the first n terms is
a (1  r )
n
Sn 
Proof:
1 r
S n  a  ar  ar  ar  ...  ar
2
3
S n r  ar  ar  ar  ...  ar
2
3
S n  S n r  a  ar
n 1
n
n
 S n (1  r )  a (1  r )
n
a (1  r )
n
 Sn 
(1  r )
(geometric series only)
Ch. 1- Infinite Series>Convergence and Divergence>Geometric Series
The sum of the geometric series is then:
a (1  r )
n
S  lim S n  lim
x 
(1  r )
x 
if r > 1, then r gets infinitely big as n   , so S  
n
if r < 1, then r gets infinitely sm all as n   , so
n
S 
a
1- r
(for |r|<1, geometric series only)
ex: 0 . 555 5 
5

10
where
5
5

100
a
2
1000
5
10
ex: 0.583333…
 ...  a  ar  ar  ...
, r 
1
10
3
Ch. 1- Infinite Series>Convergence and Divergence>Alternating Series
B. Alternating Series:
Series whose terms are alternately positive and negative.
ex:
1) 1  2  3  4  5  ...
2)  1 
1
2

1
4

1
8
 ...
• Test for converging for alternating series:
An alternating series converges if the absolute value of the terms decreases
steadily to zero. a n 1  a n and lim a n  0
n 

ex:
a
n 1
n
 1 
1
2

1
4

1
8
 ...
Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Preliminary Test
C. More general results:
There are loads of other types of series besides geometric
and alternating. So, how do we find whether a general
series converges? This is a hard problem. Here are some
simple tests (tons more exist). We’ll look at 3 tests:
1)
Preliminary Test:
If the terms of an infinite series do not tend to zero
( lim a n  0 ) , then the series diverges.
n 
Note: this test does not tell you whether the series
converges. It only weeds out wickedly divergent series.
ex:
1)
2)
1

2
3
3
1
1
3

6

3

3

1
12
4
 ...
3

1
24
 ...
Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Preliminary Test
The next tests are for convergence of series of positive terms, or for absolute
convergence of a series with either all positive or some negative terms.
defn: Say we have a series (series #1) with some negative terms. Then say we
make a new series (series #2) by taking the absolute value of each term in the
original series. If series #2 converges, then we say series #1 converges absolutely.

ex:
a
n
 1 
1
2

1
4

1
8
 ...
n 1

let

bn  1 
1
2

1
4

1
8
 ...
n 1
If ∑bn converges, then ∑an converges absolutely.
Thm: If a series converges absolutely, then it converges.
(eg, if ∑bn converges, then ∑an converges in above example.)
Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Comparison Test
2) Comparison Test:
a) Compare your series a1+a2+a3+… to a series known to
converge m1+m2+m3+….
If a n  m n for all n from some point on, then the series
a1+a2+a3+… is absolutely convergent.
b) Compare your series a1+a2+a3+… to a series known to
diverge d1+d2+d3+….
If a n  d n for all n from some point on, then the series
a1+a2+a3+… is divergent.

ex:

n 1
1
n
2
 1
1
4

1
9

1
16

1
25
 ...
does this converge?
Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Ratio Test
3) Ratio Test:
For this test, we compare an+1 to an:
in the limit of large n:
Ratio test: If p < 1, the series converges.
If p = 1, use a different test.
If p > 1, the series diverges.

ex:

n 1
2
n
n
2

ex: Harmonic Series

n 1
1
n
Ch. 1- Infinite Series>Power Series
III. Power Series
defn: A power series is of the form:


a n ( x  a )  a 0  a1 ( x  a )  a 2 ( x  a )  ...
n
2
n 1
where the coefficients an are constants.
Note: Commonly, we see power series with a=0:


a n ( x )  a 0  a 1 x  a 2 x  ...
n
2
n 1
ex:
1) x  x  x  ...
2
2) 1 
3)
1
3
1
2
x
3
x
1
6
1
4
x 
x 
2
2
1
9
1
8
x  ...
3
x  ...
3
Ch. 1- Infinite Series>Power Series>Convergence
A. Convergence of a power series depends on the values of x. m
ex:
1
3
x
1
6
x 
2
1
9
x  ... 
3
1
3n
x  ...
n
Ch. 1- Infinite Series>Power Series>Convergence
We must consider the endpoints ±1 separately: (because these points fail the ratio test)
??? keep the following ????
if x = -1:
if x = 1:
converges by alternating series test.
(harmonic series), so it diverges at x=1.
Thus, our power series converges for -1≤ x <1 and diverges otherwise.
Ch. 1- Infinite Series>Power Series>Expanding Functions
B. Expanding functions as power series:
From the previous section, we know that the sum of a power series depends on x:

S ( x) 

an ( x  a)
n
n0
So, S(x) is a function of x!
Useful trick: Try to expand a given function f(x) as a power series (Taylor series.)
(We often do this when the original function is too complex to use easily.)
ex: f(x) = ex
Ch. 1- Infinite Series>Power Series>Expanding Functions
More generally: How do we find the Taylor Series expansion of a general function f(x):
f ( x )  a 0  a1 ( x  a )  a 2 ( x  a )  a 3 ( x  a )  ...
2
3
(This approximates f(x) near the point x=a.)
Here’s how:
f ( x )  a 0  a 1 ( x  a )  a 2 ( x  a )  a 3 ( x  a )  a 4 ( x  a )  ...
2
3
4
f ' ( x )  e  a 1  2 a 2 ( x  a )  3 a 3 ( x  a )  4 a 4 ( x  a )  ...
x
2
3
f ' ' ( x )  e  2 a 2  6 a 3 ( x  a )  12 a 4 ( x  a )  ...
x
2
f ' ' ' ( x )  e  6 a 3  24 a 4 ( x  a )  ...
x
f ( x )  n! a n  ( n  1)! a n  1 ( x  a )  ( n  2 )! a n  2 ( x  a )  ...
n
1
2
Evaluating each of these at x=a:
f ( x  a )  a0  a0  f (a )
f ' ( x  a )  a1  a1  f ' ( a )
f ''( x  a)  2a2  a2 
1
2
f ' ' ' ( x  a )  6a3  a3 
1
6
f ( x  a )  n! a n  a n 
1
n!
n
f ' ' (a )
f ' ' ' (a )
n
f (a )
So, our Taylor series expansion of f(x) about the point x=a is:
f ( x )  a 0  a 1 ( x  a )  a 2 ( x  a )  a 3 ( x  a )  a 4 ( x  a )  ...
2
f(x)  f(a)  f' (a)(x  a) 
1
2!
3
2
f' ' (a)(x  a) 
4
1
3!
3
f' ' '(a)(x  a)  ... 
1
n!
n
n
f (a)(x  a)  ...
Ch. 1- Infinite Series>Power Series>Expanding Functions
defn: a MacLaurin Series is a Taylor Series with a=0.
ex: f(x) = sin(x)
ex: Electric field of a dipole
Ch. 1- Infinite Series>Power Series>Expanding Functions
And, you can do all sorts of math with these series to get other series…
(see section 13 for examples)
ex: (x2+3) sin(x) (find the MacLaurin Series expansion.)
ex: sin(x2)