MATH101 April 2014
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q2 • Q3 • Q4 • Q5 • Q6 • Q7 (a) • Q7 (b) • Q8 • Q9 (a) • Q9 (b) •
Question 01 (f)
Short Problem. Show your work. No credit will be given for the answer without the correct accompanying work.
Find the values of p for which the series converges.
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!
Consider the integral test.
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To use the integral test, we need to compute
Let , then . With this change of variables, when then and when then does as well. Applying the substitution,
Any restrictions on will come if we cannot substitute the infinite upper limit and get zero. We have the following for the indefinite integral of :
The only case in which plugging in and infinite limit will result in zero is for . Since the Right hand side of the equality converges whenever , by the integral test, the series converges for .
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