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 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
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
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- 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—$9+-
23_w_3 2“_£4_.
(13)? 32+?3 5-4'
(0) 27:0
2. Suppose Mm) = Zena” has radius of convergence R >
Math Assignment 8 due Wednesday March 20, 2013 at 4:30 pm.
[O 1. (a) Find the interval of convergence for each of the following power series and explain
What ha ens at the end oints.
pp p \méhmg ,ﬂvéaas’i
W
um) 22:1 mac + 1r 2» (iv) Enos—1r + are -« 1r 1
Math ASSlgDmeﬂt 9 due Wednesday March 27, 2013 at 4:30 pm.
4’ Z— 1. (a) Suppose 22:1 lanI < 00. Show that 2:11 on cos me converges uniformly on R.
7" (b) Prove that 27:1 mne’m‘ converges uniformly on [0, oo).
Sb 3 2. (a) Prove that 22:0 2” sin (1/ (37%) c
Math 148 Assignment 7 due Friday March 8, 2013 at 10:30 am.
i 1. Decide Whether the following series converge or diverge and state if the convergence is
only conditional.
\ 2. Suppose an 2 O and {an} is decreasing. Prove that Zen converges ifwand on
Math 148 Assignment 5 due Friday Feb. 15, 2013 at 10:30 am.
1. Evaluate the following integrals. a, b will denote non—zero constants.
(a) /(ln$)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 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
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
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 = ——
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
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. =
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
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