School

University of AlbertaDepartment

MathematicsCourse Code

MATH214Professor

Pierre YoussefStudy Guide

QuizThis

**preview**shows half of the first page. to view the full**2 pages of the document.**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(e−n). Is it convergent?

2. Is the following series convergent?

∞

X

n=1

5en+2 sin(e−n)

Problem 2. (5 pt.) Is the following series convergent? If yes, ﬁnd its sum.

∞

X

n=0

(−3)n+22−2n

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(e−n) = lim

n→∞

e2sin(e−n)

e−n=e2lim

x→0

sin(x)

x=e2, where we made the change of vari-

ables x=e−n. Therefore, this sequence is convergent.

2. Since lim

n→∞

5en+2 sin(e−n)=5e26= 0, then by the test of divergence, this series is divergent.

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