Theorem. If f is continuous on [a, b], then
f (x) dx exists.
Improper integrals violate this theorem in one of two ways:
Type 1: Integration over an unbounded interval.
Type II: Integration where the integrand becomes unbounded at
one or both endpoi
Department of Mathematical Sciences
CARNEGIE MELLON UNIVERSITY
21122 Integration, Different
Test 3: April 22, 2014
ial Equations, and Approximation
E TuTh 8:30 PH DH 1211 Chu,
F TuTh 10:30 PH 125C Cheung
G TuTh 12:30 WEH 5302 C
Theorem. For a given power series
cn (x a)n there are only three possibilities:
(i) The series converges only when x = a.
(ii) The series converges for all x.
(iii) There is a positive number R such that the series converges if
|x a| < R and diverge
Bounded sequence theorem
Every bounded monotonic sequence con
A series 2 an converges if the sequence of
partial sums dened by
Most of the tests on this summary in some
way nd an upper bound0r sho
A sequence is a function with domain a set of integers usually the positive integers or
the non-negative integers.
The domain element is indicated by a subscript as in an =
2n + 1
A sequence of the form
a, ar, ar2 , ar3 , , arn ,
Eulers method is an elementary numerical approach to determining an approximate solution
to initial value problems of the following form:
= f (x, y), y(a) = y0 , a x b
The method rests on the observation that the slope of the tangent
(i) If lim
(ii) If lim
|an | = L > 1 or lim n |an | = , then the series
an is divergent.
(iii) If lim
|an | = L < 1, then the series
an is absolutely convergent.
|an | = 1, the Root Test is inconclusive.
Example. Approximate the integral
ex dx to 2 significant figures.
Solution: We will approximate the integrand by a Taylor polynomial so that we can
integrate it! and use Taylors Theorem with remainder to justify the answer.
Substituting x2 for x i