Sequences and Series 6: Taylor series
12 videos • 12,552 views • by Jim Fowler It is time for Taylor series! Last week, in Week 5, we met power series. The big change for us was that, instead of considering a single series, last week we started considering series like sum_{n=0}^infinity a_n x^n that included a parameter x. We learned how to recognize some power series as well-known functions for certain values of x, like sum_{n=0}^infinity x^n = 1/(1-x) when |x| is less than 1. So what are Taylor series? Instead of starting with a power series and finding a nice description of the function it represents, we will start with a function, and try to find a power series for it. There is no guarantee of success! But incredibly, many of our favorite functions will have power series representations. Sometimes dreams come true. In many ways, there are many things that will be left unsaid. I hope this brief introduction to Taylor series will whet your appetite to learn more calculus. If you have any questions or concerns about the content or the course, I also hope you will post your questions or write to me at fowler@math.osu.edu. I like being able to chat about mathematics with you. Finally, this is the last week of the course. Let me tell you that it has been my honor and my pleasure to be one of your guides through mathematics. I really enjoyed putting this course together, and I very much look forward to more. I hope we will meet again.