Test 3 Review Sheet Hints—Math 122
Spring 2005, Prof. Frank
(1) True or false?
(a) False. {−1n}is bounded but not convergent.
(b) True. This is a theorem but you should be able to convince yourself of it by
drawing the graph of a convergent sequence.
(c) True, by the defintions of increasing and nondecreasing.
(d) False. The constant sequence 1,1,1,1,1, ... is nondecreasing but not increasing.
(2) A sequence is decreasing if ak> ak+1 for all k= 1,2,3, ..., whereas it is nonincreasing
if ak≥ak+1 for all k= 1,2,3, .... The sequence 0,0,0,0,... is nonincreasing but not
decreasing, and you should come up with the sketches yourself.
(3) What section were those in again?
(4) 0/0 is in section 10.5 and the rest are in section 10.6.
(5) The form 0/0 is indeterminate because a limit in this form can go to any real number,
or it can diverge.
(6) Always! If the integral is indefinite, we must think of it as the limit of definite
integrals in order to compute a value for it. We have to continue using the “lim”
while we do the integration and plug in the endpoints. After that we can take the
limit, and only after that can we stop writing “lim”.
(7) Because it is undefined at −1, Z2
−3
dx
x+ 1 is an improper integral. We would write
Z2
−3
dx
x+ 1 = lim
b→−1−Zb
−3
dx
x+ 1+ lim
a→−1+Z2
a
dx
x+ 1
You should try evaluating these integrals: you’ll find that they both diverge and so
the original integral diverges too.
(8) When you specify lim
a→1+Ze
a
dx
x√ln x, you are creating a limit of proper integrals. If
you allowed ato approach 1 from both sides, half of the integrals you create are
improper, which defeats the purpose. When you compute the integral here you get:
lim
a→1+Ze
a
dx
x√ln x= lim
a→1+Z1
ln a
du
√u= lim
a→1+h2√u
1
ln ai= lim
a→1+h2−2√ln ai= 2.
(9) The worksheet I handed out allows you to see that it converges whenever p > 1 and
diverges if p≤1.
(10) The same worksheet will let you conclude that it converges whenever p < 1 and
diverges whenever p≥1.
(11) A sequence is an infinite list of numbers; a series is a sum of an infinite list of numbers.
(12) The partial sums are S1=a1, S2=a1+a2, S3=a1+a2+a3. The general form is
SN=a1+a2+... +aN=
N
X
k=1
ak. The series is given by lim
N→∞
SN.
(13) We’ve seen collapsing, geometric, p-series, harmonic series, and alternating series.
Find the precise forms of these series in your notes or the book. While you’re at it,
figure out when/whether they converge.
