Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
Section 8.3  Trigonometric Integrals
GROUP I:
sinn x dx,
cosn x dx
METHOD: Use the following identities:
n1
1
sinn x dx = sinn1 x cos x +
n
n
cosn x dx =
GROUP II:
n1
1
cosn1 x sin x +
n
n
sinn2 x dx,
cosn2 x dx,
n2
n2
sinm x cosn x dx
METHOD:
(a) If n i
Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
10.6 The Comparison, Ratio, and Root Tests
suppose that a1 b1 , a2 b2 , a3
(a) If the bigger series
bk be series with nonnegative terms and
ak and
THEOREM (The Comparison Test): Let
k=1
b3 , . . . , ak bk , . . .
k=1
bk converges, then the smaller series
Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
10.8 Power Series
DEFINITION: If c0 , c1 , c2 , . . . are constants and x is a variable, then a series of the form
ck xk = c0 + c1 x + c2 x2 + . . . + ck xk + . . .
k=0
is called a power series in x.
EXAMPLE:
xk = 1 + x + x2 + x3 + . . .
1.
k=0
2.
k=0
x2
Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
10.1 Maclaurin and Taylor Polynomial Approximation
DEFINITION: If f can be dierentiated n times at x0 , then we dene the nth Taylor polynomial for f about x = x0 to be
pn (x) = f (x0 ) + f (x0 )(x x0 ) +
f (x0 )
f (x0 )
f (n) (x0 )
(x x0 )2 +
(x x0 )3 +
Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
10.10 Dierentiation and Integration of Power Series
THEOREM (Dierentiation of Power Series): Suppose that a function f is represented by
a power series in x x0 that has a nonzero radius of convergence R; that is,
ck (x x0 )k
f (x) =
(x0 R < x < x0 + R)
k=
Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
10.4 Innite Series
DEFINITION: An innite series is an expression that can be written in the form
uk = u1 + u2 + u3 + . . . + uk + . . .
k=1
The numbers u1 , u2 , u3 , . . . are called the terms of the series.
Consider
s1 = u1
s2 = u1 + u2
s3 = u1 + u2 + u
Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
Improper Integrals
DEFINITION: The improper integral of f over the interval [a, +) is dened as
+
f (x)dx = lim
+
a
f (x)dx.
a
In the case where the limit exists, the improper integral is said to converge, and the limit is
dened to be the value of the inte
Advanced Topics in Numerical Analysis (Numerical Optimization)
MATH GA 20112

Fall 2012
9.1 FirstOrder Dierential Equations And Applications
DEFINITION: A dierential equation is an equation involving one or more derivatives of
an unknown function:
f (x, y, y , y , . . . , y (n1) , y (n) ) = 0.
()
The order of a dierential equation is the or