565 Summary of Tests for Series
Summary of Tests for Series
Kwai Bon Chiu
P. 1
Summary of Tests for Series
Kwai Bon Chiu
P. 2
1
565 Summary of Tests for Series
Summary of Tests for Series (contd)
Kwai Bon Chiu
P. 3
2
Baruch College
Department of Mathematics
MATH 3010 SYLLABUS
ELEMENTARY CALCULUS II
Textbook or ebook: Calculus 10th Edition by Larson and Edwards, Cengage Learning Publisher. The
Webassign homework correlates with the section number and topic in the textb
564 Ratio Test and Root Test
Ratio Test & Root Test
Kwai Bon Chiu
P. 1
Ratio Test
Consider the series
a
n
1
an +1
= lim an +1
n a
n a
n
n
Let = lim
If < 1, then an converges
If > 1, then an diverges
If = 1, then test fails
Kwai Bon Chiu
P. 2
1
564 Ratio
570 Guidelines for Testing a Series
Guidelines for Testing a Series
for
Convergence or Divergence
Kwai Bon Chiu
P. 1
lim an 0 ?
n
Yes
a
n
diverges
No
a
n
is one of special types :
Geometric, p-series,
Telescoping,
Alternating ?
Yes
Use appropriate
test
Power Series I
Kwai Bon Chiu
P. 1
Power Series
an x n
Power Series is an infinite series in the form
n =0
More generally, series of the form
series centered at c.
Note :
an (x c ) is a power
n
n =0
an (x c ) = a0 + a1 (x c ) + a2 (x c ) + .
n
2
n =0
T
Power Series II
(Taylor and Maclaurin Series)
If f is represented by a power series f ( x) = an ( x c ) for
n
f ( n ) (c )
and
all x in an open interval I containing c, then an =
n!
f ' ' (c )
f ( n ) (c )
f ( x) = f (c) + f ' (c)( x c) +
( x c) 2 + . +
(
AP
www.yunzhou.org
AP
Q 207196002
VXZ:
Value of function
Variable
Vector
Velocity
Vertical asymptote
Volume
X-axis x
x-coordinate x
x-intercept x
Zero vector
Zeros of a polynomial
T:
Tangent function
Tangent line
Tangent plane
Tangent vec
Northfield Mount Hermon School
Using the TI-89 in Mathematics
This manual was created by members of the Northfield Mount Hermon mathematics department in June of 2001. It is designed to supplement the instruction of mathematics in our curriculum and to se
560 The Integral Test
The Integral Test
Kwai Bon Chiu
P. 1
The Integral Test
Suppose f is a continuous, positive, decreasing function on [1, )
and let an = f (n). Then the series an and the improper integral
n =1
1
f ( x) dx either both converge or both
561 Comparison Test
Comparison Test for Series
Kwai Bon Chiu
P. 1
Direct Comparison Test
n
the larger converges
the smaller must
converge
the smaller diverge
the larger must
diverge
These tests will greatly expand the variety of series that we ar
Alternating Series Test
Prepared by Kwai Bon Chiu
P. 1
Alternating Series
Alternating Series is a series whose terms
alternate in signs.
i.e.
(1)
n =1
Prepared by Kwai Bon Chiu
n
an or
(1)
n =1
n +1
an where an > 0
P. 2
1
Alternating Series Test
Let a
Course Policy: Fall 2015
Mathematics 3010 JMWA M/W 12:25- 14:05 Room 12-175
Professor Bruce W. Jordan
Ofﬁce 6-221
bruc (aordani’iiliharuch .c uny. ed u
(646) 3124128
0 Attendance is taken every class. The College policy is that a student is permitted to