Math 115 HW 4 Part 1

Goals:
Math 115 HW 4 Part 1
Due Wednesdy, October 31
• Get familiar working with diverging sequences.
• See how fundamental limits like lim a n = 1 for |a| < 1 and lim a n = ∞ for
|a| > 1 are used.
• Get familiar working with lim inf and lim sup
• See how the fact that monotone sequences have limits can be used for computing
limits of recursively defined sequences.
Total number of possible points: TBD.
(Optional) What is the total time you spent on this homework?
Of which working alone: Working with other students:
At office hours/tutoring: Other(specify):
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Optional: ∗ What was the most confusing thing this week? For what would
you like an explanation or clarification?
Review/fun problem:
1) Let x = 3√ 2 + 3√ 4. Show that x is irrational. † Hint: Suppose x is rational.
Then x 2 is also rational, and so is x 2 − x − 4.
2) Let y = 3√ 2 + 3√ 3. Show that y is irrational. Hint: Suppose y is rational.
Then y 3 is also rational, and so is y3 −5
3y .
Do the following problems from Ross:
9.10
9.12
9.14
9.18
10.2
10.10
∗ I will address some of the questions raised here on the class, without revealing the identity of
the source. If you would rather not have your question appear on the page, please state so before
the question.
† We haven’t discussed this in detail yet, but for any s there is unique real numbers t such that
t 3 = s and this is the number we mean when writing 3√ s. It is of course true that 3√ ab = 3√ a 3√ b
and 3√ a 3 = a.
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