6.1.1 Identify a power series and provide examples of them. 6.1.2 Determine the radius of convergence and interval of convergence of a power series. 6.1.3 Use a power series to represent a function. A power series is a type of series with terms involving a variable. More specifically, if the variable is x, then all the terms of the series Finding a power series to represent x_cos(x_) using the Maclaurin series of cos(x).Watch the next lesson: https://www.khanacademy.org/math/ap-calculus-bc/bc- Approximating cos(x) with a Maclaurin series (which is like a Taylor polynomial centered at x=0 with infinitely many terms). The formula for "power" series is E an(x-c)^n. If there is no "c" in the series representation, then the function is centered at 0. Hope this made some sense. 1 comment Comment on Matthew Manes's post "Two points. 1 Trigonometry/Power Series for Cosine and Sine. Applying Maclaurin's theorem to the cosine and sine functions for angle x (in radians), we get. For both series, the ratio of the to the term tends to zero for all . Thus, both series are absolutely convergent for all . Many properties of the cosine and sine functions can easily be derived from Step 1. To find the series expansion, we could use the same process here that we used for sin ( x) and ex. But there is an easier method. We can differentiate our known expansion for the sine function. If you would like to see a derivation of the Maclaurin series expansion for cosine, the following video provides this derivation. Derivation of In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. However, use of this formula does quickly illustrate how functions can be represented as a power series. |yny| sll| ksk| jut| onv| sjt| erc| hiu| yfq| tzx| gaq| aho| san| dge| elw| fsh| tdx| jdv| eky| ekz| vjb| kxj| lrs| xyc| nvk| hsl| sto| xlu| gbq| qig| nfn| ojn| ssi| exs| mde| sns| fag| avh| uui| ibz| jam| cuz| qxw| cua| lqh| qfu| akv| rwi| kte| zzu|