Transcript Document
Chapter 1: Infinite Series I. Background
• An infinite series is an expression of the form:
n
1
a n
1
a
2
a
3
a n
...
where there is some rule for how the a’s are related to each other.
Ch. 1- Infinite Series>Background
ex:
2) 1 2 4 1 8 ...
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?) T stop 4
s
4
s
3 4
s
6 ...
*picture?*
Ch. 1- Infinite Series>Convergence and Divergence 2) Complicated math expressions can be approximated by series and then solved more easily.
ex:
e x
(1
x
3
3
x
4
6
...)
dx
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 n
k n
1
a k a
a n S
lim
x
S n
terms of the series.
ex:
n
1 1
n
1 2 1 3 1 4 ...
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
2
ar
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
S n
a
(1
r n
) 1
r
Proof:
S n ar
ar
2
S r n S n
ar
S r n
ar
2
ar ar n
3
ar
3
S n
(1
r a
(1
r n
)
S n
a
(1
r n
) (1
r
)
ar n ar n
1 (geometric series only)
Ch. 1- Infinite Series>Convergence and Divergence>Geometric Series The sum of the geometric series is then:
S
x
lim
S n
x
lim
a
(1
r n
) (1
r
) n if r >1, then r gets infinitely big as n , so S , so
S
a
1-
r
(for |r|<1, geometric series only)
ex:
0 .
555 5
5 10
5 100
5 1000
...
a
ar
2
ar
3
...
where
a
5 10
, r
1 10
ex:
0.583333…
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. and
n
1
a n
lim
n
a n
0
ex:
n
1
a 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 (
a n
0 ) , then the series diverges.
n
lim Note: this test does not tell you whether the series converges. It only weeds out wickedly divergent series.
ex:
1 ) 1 3 2 3 3 3 4 3 ...
2 ) 1 3 1 6 1 12 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:
n
1
a n
1
let
n
1
b n
1
1 2
1 2
1 4
1 4 1 8
...
1 8
...
If ∑b n converges, then ∑a n converges absolutely.
Thm: If a series converges absolutely, then it converges. (eg, if ∑b n converges, then ∑a n converges in above example.)
Ch. 1- Infinite Series>Convergence and Divergence>More General Results>Comparison Test
2) Comparison Test: ex:
a) Compare your series a 1 +a 2 +a 3 +… to a series known to converge m
a n
m n
1 +m 2 +m 3 +….
a 1 +a 2 +a 3 +… is absolutely convergent.
b) Compare your series a 1 +a 2 +a 3 +… to a series known to diverge d
a n
d
1 +d 2 +d 3 +….
If for all n from some point on, then the series
n
a 1 +a 2 +a 3 +… is divergent.
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 a n+1 in the limit of large n: to a 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
III. Power Series defn:
A power series is of the form:
n
1
a n
(
x
a
)
n
a
0
a
1 (
x
a
)
a
2 (
x
a
) 2 ...
where the coefficients a n are constants.
Note: Commonly, we see power series with a=0:
n
1
a n
(
x
)
n
a
0
a
1
x
a
2
x
2 ...
ex:
1 )
x
x
2
x
3 ...
2 ) 1 1 2 3 ) 1 3
x
x
1 6 1 4
x
2
x
2 1 8 1 9
x
3 ...
x
3 ...
Ch. 1- Infinite Series>Power Series
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 ...
3 1
n 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: converges by alternating series test.
if x = 1: (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
)
n
0
a n
(
x
a
)
n
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) = e x
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
a
1 (
x
a
)
a
2 (
x
a
) 2
a
3 (
x
a
) 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
) 2
a
3 (
x
a
) 3
a
4 (
x
a
) 4 ...
f
' (
x
)
e x
a
1 2
a
2 (
x
a
) 3
a
3 (
x
a
) 2 4
a
4 (
x
a
) 3 ...
f
'' (
x
)
e x
2
a
2 6
a
3 (
x
a
) 12
a
4 (
x
a
) 2 ...
f
' '' (
x f n
(
x
) )
e x
6
a
3 24
a
4 (
x
a
) ...
n
!
a n
(
n
1 )!
a n
1 (
x
a
) 1 (
n
2 )!
a n
2 (
x
a
) 2 ...
Evaluating each of these at x=a:
f f f f f
(
x
' (
x
a
)
a
)
a
0
a
1
a
0
a
1 '' (
x
a
) 2
a
2
a
2 ' '' (
x
a
) 6
a
3
a
3
f f
(
a
) ' (
a
) 1 2 1 6
f f
' '' (
a
) '' (
a
)
n
(
x
a
)
n
!
a n
a n
1
n
!
f n
(
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
) 2
a
3 (
x
a
) 3
a
4 (
x
a
) 4 ...
f(x) f(a) f' (a)(x a) 1 2!
f'' (a)(x a) 2 1 3!
f''' (a)(x a) 3 ...
1 n!
f n (a)(x a) n ...
defn:
a MacLaurin Series is a Taylor Series with a=0. Ch. 1- Infinite Series>Power Series>Expanding Functions
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: (x 2 +3) sin(x) (find the MacLaurin Series expansion.) ex: sin(x 2 )