r2, 5.
Es
+303”
/
Math Assignment 1 due Friday January 18, 2013 at 10:30 am.
1. Suppose an m 1, yo = 0, and inductively deﬁne 33,1 = an“; +2yn_1 and yn = 33ml +yn_1
for n = 1,2,3, Show that mi — y: 2 :l:1 and the sequence (an/yn)ﬁ°=1 converges to
x/i.
. D
4/
WNW».
£91» R? 5%
Math 148 Assignment 3 due Friday Feb. 1, 2013 at 10:30 am.
1. (a) Find the derivatives of the following functions: W dim;
, i.
“verses? a
M‘VE’WJMQ‘E 23‘
” -(ffsin4tdt) 1 M M +1;
é ' F = _._
(I) (m) 1+ sin6 t + tﬁdt
‘15 '1’ dt
" F = ——
1. (a) The area bounded by the 5: axis, continuo
Math 148 Assignment 4 due Friday Feb. 8, 2013 at 10:30 am.
us positive function y 2 ﬁx) and the
lines a: m 1,3: 2: 6 equals Vb? —§— — x/ﬁ for all I: > 1. Find f(x) for a: > 1.
(b) Find the area of the regio
Math Assignment 6 not to be handed in.
1. (a) Show that If” m“Pdm converges if p > 1 and diverges if p g 1.
(b) Decide Whether or not the following improper integrals converge
. 00 [1x . 1 dz . no 1 m
(1) f1 m (11) f0 3; 1+3; (“0 f0 W553:
2. (a) Compu
Math 148 Assignment 5 due Friday Feb. 15, 2013 at 10:30 am.
1. Evaluate the following integrals. a, b will denote non—zero constants.
(a) /(ln$)3dm
(b) [ﬁx/Woe
(c) / arctan ﬂag;
(a) / earsinbxdw
2:1:2 +03+1
(8) fun + 3m w n26”
m3+$+2
f Wail
()/34+2x2+1$
d
Math 148 Assignment 7 due Friday March 8, 2013 at 10:30 am.
i 1. Decide Whether the following series converge or diverge and state if the convergence is
only conditional.
\ 2. Suppose an 2 O and {an} is decreasing. Prove that Zen converges ifwand on
Math ASSlgDmeﬂt 9 due Wednesday March 27, 2013 at 4:30 pm.
4’ Z— 1. (a) Suppose 22:1 lanI < 00. Show that 2:11 on cos me converges uniformly on R.
7" (b) Prove that 27:1 mne’m‘ converges uniformly on [0, oo).
Sb 3 2. (a) Prove that 22:0 2” sin (1/ (37%) c
Math Assignment 8 due Wednesday March 20, 2013 at 4:30 pm.
[O 1. (a) Find the interval of convergence for each of the following power series and explain
What ha ens at the end oints.
pp p \méhmg ,ﬂvéaas’i
W
um) 22:1 mac + 1r 2» (iv) Enos—1r + are -« 1r 1
/M
39*?"
- a (d) Find the Taylor series for (1 «~— 33—1/2
Math 148 Assignment 10 due Friday Apr 4, 2013 at 10:30 am.
1. Evaluate the following:
(a)1wm3+$5—$9+-
23_w_3 2“_£4_.
(13)? 32+?3 5-4'
(0) 27:0
2. Suppose Mm) = Zena” has radius of convergence R >
THE RADIUS OF CONVERGENCE FORMULA
A real sequence cfw_xn may have no limit, but it always has so-called upper and
lower limits, denoted
lim sup xn and lim inf xn .
Each of these can assume the extended real values + and . The notation
lim sup xn literall
Math 148 Study Notes
E. Hashman
April 20, 2011
1. Integration Theory.
Upper and Lower Sums.
Let
f
be any bounded function over a closed interval.
f
may be positive, negative, or possibly discontin-
uous.
A
partition P
of
[a, b]
is an increasing list of nu
Tests for Convergence of a Series
Original Source: facultyfiles.deanza.edu/gems/bloomroberta/ConvergenceTests.doc
Edited by Shawn Liu
Test for Divergence
Given any series an, if lim a n 0 or does not exist, then the series an is divergent.
n"!
Note that g
Math Assignment 2 due Friday January 25, 2013 at 10:30 am.
4— 1. Suppose f : [(1, 1)} «~—) R is continuous, f(:r) 2 0 for all a: and f(c) > 0 for some 0 E [a, b].
Show that f: f > 0. ' _
4 2. Show that if f , g : [(5, b} a R are both integrable, then b. =
Chapter 4
Series
Divergent series are the devil, and it is a shame to base on them
any demonstration whatsoever. (Niels Henrik Abel, 1826)
This series is divergent, therefore we may be able to do something
with it. (Oliver Heaviside, quoted by Kline)
In t
math 131
infinite series, part vii: absolute and conditional convergence
Absolute and Conditional Convergence
Sometimes series have both positive and negative terms but they are not perfectly
alternating like those in the previous section. For example
sin
University of Waterloo
Faculty of Mathematics
MATH 148 Spring 2000
Mid-Term Test
Instructor: J. Wainwright Time: 1% hours
1. Attempt all of questions 15.
Two bonus questions, worth a maximum of three marks each, will be graded.
2. Your grade will be
Winter 2017
MATH 148: Additional Problems 1
1. A cycloid is the curve generated by a point on the circumference of a wheel as the wheel roles along a straight
line. Consider a wheel rolling along the x-axis on the xy-plane. The wheel has radius a and rota
Winter 2017
MATH 148: Assignment 4
Due Tuesday, February 28 at 23:59
Your solutions should be presented in clear, concise and logical steps that
completely reveal what you did to obtain your solutions.
1. Consider a continuous function f (x) that is perio
Winter 2017
MATH 148: Additional Problems 1
1. A cycloid is the curve generated by a point on the circumference of a wheel as the wheel roles along a straight
line. Consider a wheel rolling along the x-axis on the xy-plane. The wheel has radius a and rota
Winter 2017
MATH 148: Additional Problems 3
1. Find the general solution of the following DEs.
(a)
(b)
(c)
(d)
y 0 + xy = x3
x0 (t) = x tan t + sin t
x0 = x sin t + 2te cos t
y 0 = y + ex
(e) y 0 =
(f) y 0 =
1+2y 2
x3
(g) y 0 e3x (1 2y 2 ) = 0
x2
y2
(h)
y
winter-2017-mathl4B-midterm
#138 l of 16
l WK? E R Loo
m Examination
WatIAM/Quest Login Userid:
mm: Midterm
Winter 2017
Times: Monday 201702-13 at 19:00 to 20:50 (7 to 8:50PM) MATH 148
Duration: 1 hour 50 minutes (110 minutes)
Exam ID: 3493 820
Sections
Alternating series and absolute convergence (Sect. 10.6)
I
Alternating series.
I
Absolute and conditional convergence.
I
Absolute convergence test.
I
Few examples.
Alternating series
Definition
An infinite series
P
an is an alternating series i holds eith
1
Sample Solutions for Midterm I review problems
Math 451, Section 3, Fall 2014
11.3) a) (s6n ) = (1) is a monotone subsequence of (sn ). (tn ) is decreasing so any subsequence
1
of (tn ) is decreasing, hence monotone. (u2n ) = ( 22n
) is a decreasing sub
86
2.7.
C HAPTER 2
I NFINITE S ERIES
T HE A LTERNATING S ERIES T EST
We have focused almost exclusively on series with positive terms up to this point. In
this short section we begin to delve into series with both positive and negative terms,
presenting a
Math 142, Exam 3 Information.
4/14/11, LC 113, 9:30 - 10:45.
Exam 3 will be based on:
Sections 11.3 - 11.8;
The corresponding assigned homework problems
(see http:/www.math.sc.edu/boylan/SCCourses/142Fa10/142.html)
At minimum, you need to understand how
Math 3001
Due Wed Oct 19
Assignment #5
P1
P1
1. Given the series
n=1 an and
n=1 bn , suppose that there
P1exists a
number N such
that
a
=
b
for
all
n
>
N
.
Prove
that
n
n
n=1 an is
P1
convergent i n=1 bn is convergent.
P
P1
Is it true that 1
n=1 an =
n=1
smi98485_ch08b.qxd
666
5/17/01
Chapter 8
2:07 PM
Page 666
Infinite Series
8.5
ABSOLUTE CONVERGENCE
AND THE RATIO TEST
You should note that, outside of the Alternating Series Test presented in section 8.4, our
other tests for convergence of series (i.e., t