Hi all,

We have been looking at the power series for the whole week last week! The idea is that if the function is “nice” enough, we can represent it as the infinite sum of powers of x. Some functions are hard to study (for example, there is no way to integrate e^{-x^2} with FTC), but we know how to integrate, differentiate polynomials very easily. A lot of properties can be obtained just by looking at the powers of x, for example, we have seen that from the powers, we can tell sine is an odd function, but cosine is an even function.

*Important fact*: power series expansions in general only work locally, so it is very important that whenever we write down a power series, we immediately check on which interval the series converges. Such an interval is always centered at the center of the series, in the form of (a-R,a+R), here we do not study the convergence at the boundary. Make sure to test for the values of x where the series converge!!

We have looked at two kinds of power series: one for functions based on \sum^{\infty}_{n=0}x^n=1/(1-x) for |x|<1, and the more general Taylor series. For both kinds, remember they are just power series, so all the facts we learned about power series apply to these series, including the differentiation and integration rules. We mostly focus on how to write the functions in power series expansions and when they converge, but we will also look at some applications of these power series.

Next week, we will wrap up the Taylor series and review for the rest of the week.

Quiz corrections are due Friday, Dec 11, before 10am. Let me know if you have any questions!

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