Hope you all had a fun and full Thanksgiving! For the past one and half weeks, we have started looking at series, whether they diverge or converge, how to compute their sums and such. For the most part, we are interested in power series and when they can be treated as functions. Next week we will continue with power series in reverse, in the sense that we look at normal functions, and see how to represent them with power series.
Series’ are sums of infinitely many items. Like how we always deal with infinities, we define series to be the limit of the partial sum. If the limit exists, we say the series converges; otherwise diverges. In this regard, we can treat the partial sums as a sequence, and all the results we had for sequences apply here. This method (looking at partial sums) normally require us to know a compact form of the partial sum. An example is the geometric series, with the compact form 1+r+r^2+…+r^n=(1-r^n)/(1-r), for all |r|>=1, the series would diverge and only for |r|<1 will the series converge.
In general, it is hard to tell if the series converges without a compact form of the partial sum, but we can tell if they diverge by looking at the limit of the term: the series will diverge for sure if the terms don’t converge to zero. You can use this standard to weed out a lot of divergent series.
In power series, we are adding polynomials of the shape c_n(x-a)^n. Note here x is the variable which is independent of the index n. As long as the series converge, we can look at its sum as a function. The values of x at which the series converge is naturally the domain of the function, therefore we focus on where the series converge.
If we are dealing with a geometric power series, the series converges if |r(x)|<1 (as in the power series). If we are dealing with a more general series, we can use the root test or the ratio test. Both tests are inconclusive at the boundary, so we omit the boundary cases(see sections 11.3 and 11.5).
For next week, we will write general functions in power series. The idea behind this is super cool: functions might be of different shapes and forms, but you can “always” write them in terms of the sum of powers of x. Therefore, if you are studying a “ugly” function, your life will be easier if you instead study the power expansion.
Alright, that is it for now. There are so many cool facts about power expansions and I can’t wait to share them with you guys! See you soon!