Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Lecture Notes on Power Series - Calculus II | MATH 1920, Study notes of Calculus

Material Type: Notes; Class: Calculus II; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

koofers-user-ktd
koofers-user-ktd 🇺🇸

10 documents

1 / 1

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
Power Series
A power series is an infinite series of the form
n0
cnxnc0c1xc2x2c3x3cnxn
where xis a variable is called a power series. More generally, series of the form
n0
cnxanc0c1xac2xa2cnxan
is called a power series centered at aor the power series about a,whereais a constant.
Note: to simplify notation we let xa01 even when xa. For a power series centered at
athere are only THREE possibilities:
(1) The series converges only when xa.
(2) The series converges for all x.
(3) There is a positive real number Rsuch that the series converges if |xa|Rand
diverges if |xa|R.
The number Ris called the radius of convergence of the power series. In case (1), R0.
In case (2), R. The set of all values of xfor which the power series converges is called the
interval of convergence of the power series.
We use the Ratio Test to find the radius of convergence of a power series. For a power
series centered about awith a finite radius of convergence R0, we must determine whether
or not the series converges at the endpoints of the interval of convergence (i.e. at aR&
aR).
Find the interval of convergence for the following power series.
1.
n0
n!xn2.
n0
1nx2n1
2n1!3.
n1
xn
n4.
n0
1nx1n
2n5.
n1
x2n
n2
Term-by-term differentiation and integration of functions defined by power series:
Let the power series
n0
cnxanhave radius of convergence R0. Then the function
fx
n0
cnxanc0c1xac2xa2c3xa3
is differentiable (and therefore continuous) on the interval aR,aRand
(a)fx
n0
ncnxan1c12c2xa3c3xa2
(b)fxdx C
n0
cnxan1
n1Cc0xac1xa2
2c2xa3
3
The series in (a) and (b) have radius of convergence R. Convergence or divergence at the
endpoints of the interval of convergence may differ from the original power series.
Find fxand fxdx and their corresponding intervals of convergence if fx
n1
xn
n.
If gx
n1
xn
n!find gx. Do you recognize this function?

Partial preview of the text

Download Lecture Notes on Power Series - Calculus II | MATH 1920 and more Study notes Calculus in PDF only on Docsity!

Power Series

A power series is an infinite series of the form

n 0

c n

x

n

 c 0

 c 1

x  c 2

x

2

 c 3

x

3

   c n

x

n

where x is a variable is called a power series. More generally, series of the form

n 0

c n

x − a

n

 c 0

 c 1

x − a  c 2

x − a

2

   c n

x − a

n

is called a power series centered at a or the power series about a, where a is a constant.

Note: to simplify notation we let 

x − a 

0

 1 even when x  a. For a power series centered at

a there are only THREE possibilities:

( 1 ) The series converges only when x  a.

( 2 ) The series converges for all x.

( 3 ) There is a positive real number R such that the series converges if |x − a |

 R and

diverges if |x − a|  R.

The number R is called the radius of convergence of the power series. In case (1), R  0.

In case (2), R  . The set of all values of x for which the power series converges is called the

interval of convergence of the power series.

We use the Ratio Test to find the radius of convergence of a power series. For a power

series centered about a with a finite radius of convergence R  0, we must determine whether

or not the series converges at the endpoints of the interval of convergence (i.e. at a − R &

a  R).

Find the interval of convergence for the following power series.

n 0

n!x

n

n 0

n

x

2 n 1

 2 n  1 !

n 1

x

n

n

n 0

n

x  1 

n

n

n 1

x − 2 

n

n

2

Term - by - term differentiation and integration of functions defined by power series :

Let the power series ∑

n 0

c n

x − a

n

have radius of convergence R  0. Then the function

f 

x 

n 0

c

n

x − a

n

 c

0

 c

1

x − a  c

2

x − a

2

 c

3

x − a

3

is differentiable (and therefore continuous) on the interval a − R, a  R and

( a ) f

x  ∑

n 0

nc n

x − a

n− 1

 c 1

 2 c 2

x − a  3 c 3

x − a

2

( b )  fx dx  C  ∑

n 0

c n

x − a 

n 1

n  1

 C  c 0

x − a  c 1

x − a 

2

 c 2

x − a 

3

The series in (a) and (b) have radius of convergence R. Convergence or divergence at the

endpoints of the interval of convergence may differ from the original power series.

Find f

x and  fx dx and their corresponding intervals of convergence if fx  ∑

n 1

x

n

n

If gx  ∑

n 1

x

n

n!

find g

x. Do you recognize this function?