Test 4 review self-tests and links (updated and with more links!)

Test 4 is scheduled for the first hour or so of class on Wednesday 6 December.

The topics are listed in the Self-Tests sheet, and also in the links below. But make sure that you practice, don’t just read or watch videos or watch someone else work problems!

This test is shorter than the previous tests, so it will include a problem of determining whether a series converges using the various comparison tests, and finding its sum also (similar to the parts of problem 3 in Test 3). You may do that problem or you may skip it as you choose. If you do it and receive a higher score than you did on any one part of problem 3 of Test 3, the higher score will be credited to your Test 3 score. (See the Test 3 review and solutions to Test 3 here.)




MAT1575Test4Review (contains two self-tests – Working now!)

Answers and links below the fold…

Sorry I couldn’t upload the document with these answers, but at least they are all working here now. Please let me know (by posting to Piazza) if you find any errors in these!


Self-Test 1

1a) The series converges by the Alternating series test: $latex\frac{1}{2n+1} is a decreasing sequence and the limit as n\rightarrow\infty is 0.

1b) The convergence is conditional: you can show that the series does not converge absolutely by limit comparison with the harmonic series, or integral comparison. (The Ratio Test fails here.)

1c) The error is less than |a_{51}| = \frac{1}{2(51)+1} \approx 0.0097 [corrected]

1d) To have the error less than  0.000005 we would need to find n such that |a_{n}| \le 0.000005, so \frac{1}{2n+1} \le 0.000005.

Solve this to find that n must be at least 100,000

2) The series converges in the interval (1, 3] Don’t forget to check the endpoints!

3) The series converges, by the Root Test

4) Substitute -x^{2} in place of x in the maclaurin series for e^{x} and simplify: e^{-x^{2}} = \displaystyle\sum_{n=0}^{\infty} \frac{(-1)^{n}x^{2n}}{n!}

5) \displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}(x-2)^{n}}{2^{n+1}}; the radius of convergence is 2, by the Ratio Test.

(Note on problem #5: you may be interested to see what happens to this series when you substitute in x=1 or x=3: it becomes a geometric series and you can check that the sum is 1 in the first case and 1/3 in the second case, as it should be. Now try substituting in x=2, and you may be surprised by what you see!)


Self-Test 2

1a) The series converges by the Alternating series test (or you could use another test to prove it converges absolutely, jumping to part 1b already. It is enough to prove absolute convergence.)

1b) The series converges absolutely: the absolute values of the terms form a p-series with p=4.

1c) The error is less than |a_{9} = \frac{1}{9^{4}} \approx 0.000152

1d) Solve \frac{1}{n^{4}} \le 0.00005 to find that n must be at least 12.

2) The series converges in the interval [2,4]. Don’t forget to check the endpoints!

3) The series converges, by the Root Test.

4) Substitute -x in place of x in the maclaurin series for e^{x}, multiply by x^{2},  and simplify: x^{2}e^{-x} = \displaystyle\sum_{n=0}^{\infty} \frac{(-1)^{n}x^{n+2}}{n!}

5) The easiest way to find this one is to take the series for \frac{1}{1-x} (the geometric series) and substitute in 3x in place of x.


The radius of convergence is \frac{1}{3}, by the Ratio Test or the Root Test.


See below for sources to help you if you are stuck on these, or of course you can post to Piazza!

Links to Paul’s Online Notes, and a few other sources:

Special Series(geometric series, p-series)

Integral comparison test

Comparison test and Limit comparison test

Strategy for series (tips on how to decide which test to use) here also is my slideshow: MAT1575Testing-strategy-for-series-&-Power-series-slideshow

and see below for a video with examples from PatrickJMT

The Alternating Series Test

Absolute convergence

Estimating the sum of an alternating series: see Example 3 here

And here is a video from PatrickJMT:  PatrickJMT on estimating alternating series

PatrickJMT on strategy for testing series including examples

(Note: This is good to review because, for testing the endpoints of the interval of convergence for a power series, we need to use every tool we have available!)