Python Programming:
An Introduction to
Computer Science
Chapter 4
Objects and Graphics
Python Programming, 2/e
1
Objectives
To understand the concept of
objects and how they can be used
to simplify programs.
To be familiar with the various
objects availab
Summary of Convergence Tests for Innite Series
Test
Series
Converges if
Diverges if
Comment
lim an = 0
cannot be used to show convergence
Divergence
an
n
n=1
arn
|r| < 1
|r| 1
1
np
n=1
p>1
Sum: S =
n=0
p-Series
Integral
f (x) dx
an
an = f (n) 0
and
n=1
bn
Review Practice Problems
1) Show that
2) Find limn
3) Are
4) Is
1
n=0 1+n2
is convergent.
(6n2 +2n+4)5
.
n10
2n n2 +6n
n=0 5+3n n2
4+5n
n=0 3+6n+1
and
n4 +2n2
n5 +6
convergent or divergent ?
convergent or divergent ?
2n
n=1 n (x
5) Find the radius of co
Chapter 1
Innite and Power series
In ancient times, there was a sense of mystery, that an innite sequence of
numbers could sum to be a nite number. Consider uniform motion that
proceeds along the x-axis from zero to one with unit velocity. The time it
tak
Summary of Convergence and Divergence Tests for Series
TEST
nth-term
Geometric
series
p-series
SERIES
an
ar
(i) Converges with sum S =
n 1
n =1
Useful for the comparison tests
if the nth term an of a series is
similar to arn-1
a
if r < 1
1 r
(i) Converg
Math 252 Fall 2002 Supplement on Eulers Method
Introduction. The textbook seems overly enthusiastic about Eulers method. These notes aim to present a more realistic treatment of the value of the method and its relation to other numerical methods for solvi
Math 252 Fall 2002 Introduction to First-Order Differential Equations
A differential equation is just an equation which involves differentials, that is to say, derivatives. A simple example is dy = 0, dt where we understand that y is a function of an inde
Math 252 Fall 2002 Some comments on bifurcations
Background. This is a slightly modied version of the notes posted on the same subject posted on the original Math 252 web page and borrowed from the UTEP SOS math project. Links to these resources are avail