calculus 2

Math 126 Name (Print): Block 3 Take-Home Due: 04/28/2017

Read Directions Entirely. By submitting this exam for a grade, you are agreeing to everything contained within.

This assignment is entirely optional. Your current score for Block 3 is: A = (Score on Test). Let B = (Score on this assignment). Your new score for Block 3 will be calculated as: Block 2 Score = A + .A(B−A

2 ).

Ex. If your test says 60, then A = 60. If you make a 100 on this assignment, then your Block 2 score is given by: Block 2 Score = 60 + .60( 100−60

2 ) = 72.

Play around with some number to see if you think it is worth your time to complete the assignment. It will be significantly harder than the in-class exam, but you have a week to complete it.

Feel free to use any sources you would like other than help from another person (discussions about how to approach a problem are fine, but you need to write out your own solutions from scratch). If the contents of your test contain work that leads me believe that you were directly and excessively working with other people, then I will submit it for academic misconduct.

Because this is a take-home assessment, the presentation of your solutions will be graded more strictly. You must show all work on each problem. You must write neatly, and the work must flow in a clear unambiguous manner, with more rigor than an in-class test. If your handwriting is terrible, you have the options of writing carefully or attaching a typed version of your solutions. I reserve the right to take away points simply because it looks sloppy or if the math syntax doesn’t quite make sense, even if I can guess what you mean (typically during in-class exams a little benefit of the doubt is given, due to the limited amount of time).

Grading will be done in ten point increments, i.e. 100, 90, 80, etc.

You must submit a stapled, printed copy to me by no later than the end of class on Friday, April 28, 2017.

Do not write graded material on this page.

Math 126 Block 3 Take-Home - Page 2 of 7 Due: 04/28/2017

1. Cauchy Condensation Test If ak ≥ 0 for all k, with ak decreasing and ak = f(k), then ∞∑ k=N

f(k) and ∞∑ k=N

2kf(2k) both converge or both diverge.

Use the Cauchy Condensation Test to prove that:

(a) ∞∑ k=2

1

k ln(k) diverges.

(b) ∞∑ k=2

1

k[ln(k)]2 converges.

Math 126 Block 3 Take-Home - Page 3 of 7 Due: 04/28/2017

2. (a) Prove whether

{ √

7, √√

7,

√√√ 7, ...

} converges or diverges. If it converges, find the

limit.

(b) Prove whether {cos(kπ 4 x)}∞k=0 converges or diverges. If it converges, find the limit.

Math 126 Block 3 Take-Home - Page 4 of 7 Due: 04/28/2017

3. (a) Suppose that ∞∑ k=2

1

(1 + c)k−3 =

16

3 for c > 0. Find the value for the constant c.

(b) Find the sum of ∞∑ k=3

12

k(k + 1) .

Math 126 Block 3 Take-Home - Page 5 of 7 Due: 04/28/2017

4. Prove whether the following series are absolutely convergent, conditionally convergent, or di- vergent.

(a) ∞∑ k=1

(−1)kk k2 + 1

(b) ∞∑ k=1

(−1)k+19k

(3k)!

(c) ∞∑ k=1

(−1)k(4k + 3) ln(2k + 1)

Math 126 Block 3 Take-Home - Page 6 of 7 Due: 04/28/2017

5. Find the interval and radius of convergence for ∞∑ k=0

(−1)k (

k

4k + 7

)k x2k.

Math 126 Block 3 Take-Home - Page 7 of 7 Due: 04/28/2017

6. Find a Taylor series for the function f(x) = 2 −x x− 1

+ (2−x)7 sin(2−x) and state the appropriate interval of convergence.