Wednesday 15 November class (Updated)

(After Test 3)

Reminder: we have a theorem about series that says: if \displaystyle\sum a_{n} converges, then \displaystyle\lim_{n\rightarrow\infty}a_{n} = 0.

So if \displaystyle\lim_{n\rightarrow\infty}a_{n} \neq 0, the series will diverge!

 

Reminder: the Integral Comparison, Direct Comparison , and Limit Comparison tests ONLY apply to series whose terms are all positive. They cannot be used on alternating series or series which have an infinite number of negative terms.

 

Absolute convergence: a series \displaystyle\sum a_{n} converges absolutely (is absolutely convergent) iff the series \displaystyle\sum|a_{n}| converges.

 

If \displaystyle\sum a_{n} converges, but \displaystyle\sum|a_{n}| diverges, we say the series is conditionally convergent.

 

Theorem: If the series is absolutely convergent, then it is conditionally convergent.

(But not the converse!)

 

Examples:

\displaystyle\sum_{n=1}^{\infty} (-1)^{n+1}\frac{1}{n} is not absolutely convergent. It is only conditionally convergent.

 

\displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}|\sin(n)|}{n^{2}} is absolutely convergent:

\displaystyle\sum_{n=1}^{\infty} \frac{|\sin(n)|}{n^{2}} converges by comparison with the p-series with p=2.

Since \displaystyle\sum_{n=1}^{\infty} \frac{(-1)^{n+1}|\sin(n)|}{n^{2}} is absolutely convergent, it converges (conditionally) as well.

 

\displaystyle\sum_{n=1}^{\infty} \frac{\sin(n)}{n^{3}} (not an alternating series) – no possibility of using the alternating series test here.

But \displaystyle\sum_{n=1}^{\infty} \frac{|\sin(n)|}{n^{3}} converges. so \displaystyle\sum_{n=1}^{\infty} \frac{\sin(n)}{n^{3}} converges absolutely, and also (therefore) converges conditionally.

 

Ratio/Root tests slideshow

MAT1575Ratio-RootTests-slideshow

Example: (My version of Problem 1 in the WeBWorK)

Consider the series \displaystyle\sum_{n=1}^{\infty}\frac{10^{n}}{(n+1)4^{2n+1}}

Evaluate the limit

\displaystyle\lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_{n}}\right|

I’ve worked this out in a separate pdf, because WordPress will not display it correctly for unknown reasons.

MAT1575WeBWorKRatioTestProblem1

 

Homework:

• Review the examples discussed in class (most of the notes are above). Make sure that you know when you can and cannot use the comparison tests (which are generally the easiest to use in the cases when they can be used).

• Do/finish the WeBWorK on Alternating Series and do the WeBWorK on the Ratio Test and Root test, but skip the problems #7, 9, 10, 11, and 14 (which refer to the root test) for now.

Note: None of the problems in the Alternating Series homework require the ratio or root tests, even the ones that ask about absolute convergence. You should try to use the Alternating Series test to test for convergence: if the problem asks about absolute convergence, try using one of the comparison tests on the series of absolute values.

Also, for the questions about estimation, you will be using the theorem from last time about the error term. Here are some videos also about this:

PatrickJMT Alternating Series Estimation Theorem

Khan Academy

• I would like to see some of the Alternating Series and Ratio Test problems on the board next time.

• No quiz next class.

• The Test 3 solutions and Post-Test strategic thinking survey are in this separate post. It is very valuable to fill out the post-test survey even if you did not fill out the pre-test survey!

 

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!