MATH 214 Quiz: Quiz 1.pdf
SchoolUniversity of Alberta
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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?
Problem 2. (5 pt.) Is the following series convergent? If yes, ﬁnd its sum.
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
Explain your solution! Write the answer! Answer is not written – problem is not solved!
en+2 sin(e−n) = lim
x=e2, where we made the change of vari-
ables x=e−n. Therefore, this sequence is convergent.
2. Since lim
5en+2 sin(e−n)=5e26= 0, then by the test of divergence, this series is divergent.
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