Monday 6 November class

Topics:

ā€¢ Some important types of series:

Geometric series

p-series

Telescoping series

ā€¢ Another theorem about convergent series:

If a series converges, then the limit of its terms must be 0.

Therefore if the limit of the terms is not 0, the series does not converge.

Note: when using theorems about series, please be very careful to distinguish the terms of the series from the partial sums of the series!

ā€¢ Intergal comparison test for convergence of a series:

Consider a series \sum_{n=1}^{\infty}a_{n} whose terms are related to a function a(x) defined and positive on the interval [1,\infty), so that a(n) = a_{n}. ThenĀ \sum_{n=1}^{\infty}a_{n} converges if and only if the improper integralĀ \int_{1}^{\infty}a(x)\textrm{d}x converges.

Example: we can use this to see which p-series will converge: the p-seriesĀ \dislpaystyle\sum_{n=1}^{\infty}\frac{1}{n^{p}} can be compared to the integralĀ \displaystyle\int_{1}^{\infty}\frac{1}{x^{p}}\textrm{d}x, and we know from our previous work that those integrals converge when p>1 and diverge when p\le1.

 

Homework:

ā€¢ Review the examples discussed in class. It is important to be able to recognize the geometric series, p-series, and telescoping series, and know who to handle them. See Section 8.2 in the textbook.

ā€¢ Do the WeBWorK: it is not due until Sunday, but your should definitely try to complete Intro to series 2 by Wednesday.

ā€¢ There will be a Quiz on Wednesday: the topic will be these types of series!

Also don’t forget that if you did not receive full scores on Problems 5 and/or 6 on Test 2, you have the opportunity to do a similar problem for half of the missing points. These problems must be written up completely correctly and are due at the start of class on Wednesday. See this post for more information and for the problems themselves. They are absolutely no excuses due on Wednesday, so don’t delay!

 

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!