Practice problems: sequences and series
For each of the following, determine whether it converges or diverges.
n
n=1 n2 +1
1.
Solution.
Comparison test. We have
1
n=1 n n
1
n n
=
n
n2
n
n2 +1
converges (it is a p-series with p = 3/2) so does
0. Since
n
n
MAT 1322 Winter 2013
DGD 10 (March 18 19th )
1. Find the Taylor series for f (x) centered at the given value of a.
a) f (x) = ln x,
a=2
b) f (x) = cos x,
a=
2. Use a Maclaurin series (Table 1, P. 613) to obtain the Maclaurin series for the given
function.
7.5 #7 The population of the world was about 5.3 billion in 1990. Birth rates in the
1990s ranged from 35 to 40 million per year and death rates ranged from 15 to 20 million per
year. Let us assume that the carrying capacity for world human population is
Lab 4
STAT5504F
Jason Ng (ID: 101024146)
Task 1:
For each generated process, plot the associated sample ACF and
PACF. Describe the characteristics of these plots.
(1)AR: X[t] = 0.4 X[t-1] + 0.6 X[t-2] + -0.3 X[t-3] + Z[t]
Description:
The ACF of AR have n
Calculus 120
Final Exam
Question 1.
a) Find the following integral:
sin ( ax ) dx
where
a is a constant, a R .
b) Find the following integral:
sin ( 8 x ) cos ( 5 x ) dx
Hint: Use part a) when solving part b). You may also want to use one of the famous
Lab 3
STAT5504F
Jason Ng (ID: 101024146)
Question 1:
In the lectures for Topic 4, I showed examples of an MA(2)
process and an MA(4) process as well as their correlograms. The
code (ma.R) for generating these TS and plotting their
correlograms is posted.
MAT1320A DGD Quiz November 4, 2016
Write your NAME and STUDENT NUMBER on a blank piece of paper, and answer the
following question. No books, notes, calculators or other devices. Show all your work and
write clearly.
(a) Find the following indefinite inte
Partial solutions for the suggested exercises (plus a
small list for review)
(1) Find the following integrals:
x2
dx.
x+1
Long division gives you (x 1 +
Integrating, we get
x2
2
1
)
x+1
dx.
x + ln(x + 1) + C.
3x3 x2 + 6x 4
dx.
(x2 + 1)(x2 + 2)
degree of
Practice problems: Maclaurin series
For each of the following functions, express it as a powerseries.
1. f (x) =
3
12x
Solution. Use
1
1x
=
n=1
xn . Replace x by 2x and multiply by 3:
3
=
3(2x)n =
3 2 n xn .
1 2x n=0
n=0
2. f (x) =
1
2x
Solution. Use
1
1x
Quiz solutions: Integration Techniques
Note that there may be other solutions possible. You can always check your
solution by dierentiating.
1.
3x2
3x3 dx
Solution. Recognize the derivative of the denominator in the numerator.
Use the substitution u = g(x