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MATH 214 Quiz: Quiz 1.pdf


Department
Mathematics
Course Code
MATH214
Professor
Pierre Youssef
Study Guide
Quiz

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Quiz #1.
First Name: Family Name: ID Number:
Instructions: Show your work, answers should be explained. If you have time, check your solution.
Do not speak to or communicate with other students. Calculators, textbooks, notes are not allowed.
Problem 1. (5 pt.)
1. Calculate the limit of the sequence an=en+2 sin(en). Is it convergent?
2. Is the following series convergent?
X
n=1
5en+2 sin(en)
Problem 2. (5 pt.) Is the following series convergent? If yes, find its sum.
X
n=0
(3)n+222n
Bonus Problem. (2 pt.) I have a 2 Liters capacity empty bottle of water. The first day, I filled
it with one liter of water. Everyday, I am allowed to add half the quantity filled the day before.
For example, the second day I would add half a liter, the third day 1/4 of a liter, etc.. I am the
only one who can add water to the bottle.
After some reflexion, I thought “I won’t be able to fill the bottle with exactly 2 Liters before I
die”. Am I right? If not, after how many days the bottle will be full containing exactly 2 liters of
water?
Explain your solution! Write the answer! Answer is not written – problem is not solved!
Solutions
Problem 1.
1. lim
n→∞
en+2 sin(en) = lim
n→∞
e2sin(en)
en=e2lim
x0
sin(x)
x=e2, where we made the change of vari-
ables x=en. Therefore, this sequence is convergent.
2. Since lim
n→∞
5en+2 sin(en)=5e26= 0, then by the test of divergence, this series is divergent.
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