MATH220 December 2009
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q2 (a) • Q2 (b) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q7 (a) • Q7 (b) • Q8 (a) • Q8 (b) • Q9 (a) • Q9 (b) •
Question 07 (b)
Decide whether the following sequence converges or diverges. Prove your answers using the definition of convergence.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
First, justify to yourself that this sequence converges to 3. Then use the definition directly to show this.
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We show that the sequence converges to 3. To see this, it suffices to show that for any there exsists an such that
for all . Let . Simplifying the left hand side above yields:
Now, choose large enough so that which can be done by choosing larger than . Then, for each , we have that
which shows that our original sequence converges to .
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