Real Analysis - Sample Midterm Help

In summary: Sorry to have mislead you for 2(c), I didn't realize that that n exponent was affecting the whole argument and not just the denominator.
  • #1
steelphantom
159
0
My professor has posted a sample midterm on her web site, but although she promised to post the solutions as well, she hasn't yet and I don't really expect her to at this point since the midterm is tomorrow. I have a few questions about some of the problems on the midterm. The sample exam can be found here: http://www.math.psu.edu/li/math312/hw.html

2(c): [tex]\sum[/tex] (2 / (3 - (-1)n))n
This series diverges, but how would I go about proving it? The first thing I thought of was the root test, but the limit of the series does not exist. What other way could I try to do this?

3: Consider a nonnegative series [tex]\sum[/tex]an. If there exists an M such that AN = [tex]\sum_{n=1}^N[/tex]an <= M for all N, then [tex]\sum[/tex]an is convergent.
This doesn't say to explicitly prove this, but I assume that's what I need to do. Isn't this true from the definition of convergence of infinite series?

For problems 6-8, it says "Find the limit of xxx." I can just stare at them and figure out what the limit is, but that's obviously not what my professor wants me to do. I guess the only thing I could do is say what the limit is and then prove it using the definition of a limit.

I know I should have asked my prof personally about this stuff, but I couldn't make her office hours yesterday, and the next office hours are Thursday, which will do me no good. As always, thanks for any help.
 
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  • #2
For 2(c), notice that 3-(-1)^n is always bigger than 2, so (2 / (3 - (-1)n))n is always less than 2/2^n

3: It indeed follows from the definition. You are told that the partial sums are bounded by M... and recall that if a sequence is monotone and bounded, then it converges. :-o

for 6-8, this is exactly what you must do... figure out what your intuition tells you about the limit to those things, then prove using the dfn of limit that your intuition is right
 
  • #3
Thanks for your help, quasar! But for 2(C), since I'm trying to prove the series diverges, don't I want to find a series that is smaller than it, rather than larger than it, if I want to try to use the comparison test?
 
  • #4
steelphantom said:
Thanks for your help, quasar! But for 2(C), since I'm trying to prove the series diverges, don't I want to find a series that is smaller than it, rather than larger than it, if I want to try to use the comparison test?

In order for a series to converge, the limit of the terms must be zero. This series fails that test since every other term is 1.
 
  • #5
NateTG said:
In order for a series to converge, the limit of the terms must be zero. This series fails that test since every other term is 1.

Well that was easy! :rolleyes: Thanks for pointing that out.
 
  • #6
I'm tagging this question on the end, even though I really should create a separate topic instead:

Would restudyng the first 2 years-worth of undergraduate Calculus several times be helpful for later studying Real Analysis?
 
  • #7
Sorry to have mislead you for 2(c), I didn't realize that that n exponent was affecting the whole argument and not just the denominator.

For your question, I would be inclined to answer "no". Well yes, it would certainly be helpful, how can it hurt? But there are better ways to use your time.
 

Related to Real Analysis - Sample Midterm Help

1. What is real analysis and why is it important?

Real analysis is a branch of mathematics that deals with the study of real numbers and their properties. It is important because it provides the foundation for many other areas of mathematics and has numerous applications in fields such as physics, engineering, and economics.

2. What topics are typically covered in a real analysis course?

A typical real analysis course will cover topics such as sequences and series, continuity, differentiability, integration, and convergence of functions. It will also introduce students to the concept of proofs and their use in mathematical arguments.

3. What are some common challenges students face when studying real analysis?

Some common challenges students face when studying real analysis include understanding the abstract concepts and techniques, developing strong proof-writing skills, and being able to visualize and apply the concepts to real-world problems.

4. How can I prepare for a real analysis midterm?

To prepare for a real analysis midterm, it is important to review all the material covered in class, practice solving problems and proofs, and seek help from the instructor or a tutor if needed. It is also helpful to work through previous exams or practice problems to familiarize yourself with the types of questions that may be asked.

5. What are some tips for success in a real analysis course?

Some tips for success in a real analysis course include attending all classes and taking good notes, actively participating in class discussions and asking questions, practicing regularly and seeking help when needed, and studying consistently rather than cramming before exams. It is also helpful to form study groups with classmates to review and discuss course material.

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