# CAS MA 124 Study Guide - Midterm Guide: Improper Integral, Absolute Convergence

150 views2 pages

10 Oct 2018

School

Department

Course

Professor

MA 124 - Spring ’10 - Exam 2 - Prof. Meuser

1) (13 points) Consider the deﬁnite integral

Z0

−2

1

√x+ 2 dx

a) Is this integral improper or not ? Explain your answer.

b) Determine whether this integral converges or diverges. Explain your answer using

properties of integrals and limits.

2) (14 points) Determine whether the improper integral

Z∞

0

xe−xdx

converges or diverges. Write any necessary limits or theorems needed to determine your

answer. For any limits you are ﬁnding, explain why those limits exist or do not exist using

properties of limits.

3) (14 points) Determine whether the following sequences converge or diverge. If they

converge ﬁnd the limit. Explain your answers using properties of limits.

a)an= (−1)nn

(n+ 1)2b)an=n2e−n

4) (14 points ) Consider the number

0.73 = 0.73737373 · · ·

a) Write this number in terms of a geometric series.

b) Does this geometric series converge or diverge ? Explain your answer.

c) If this geometric series converges ﬁnd it’s sum.

c) Express this number as a ratio of integers, i.e. as a fraction a/b for integers aand b.

5) (13 points) Determine whether the following series converges or diverges

∞

X

n=1

ne−n

6) (13 points) Determine whether the following series converges or diverges

∞

X

n=1

n3

n4−1

QUESTION 7 IS ON THE BACK OF THIS PAGE