Lecture 19: Taylor series Calculus II, section 3 April 20, 2022 Last time, we introduced Taylor series to represent (reasonably) arbitrary functions as power series, looked at some examples (around di erent points and with di erent radii of convergence), and as an application proved Euler's formula, which we used extensively to first few terms of the Taylor series of f about x = 0. Since Eq. (4) and Eq. (2) give polynomial representations of these functions, a natural guess (whose proof I leave to you if you are mathematically inclined) is that if we simply substitute Eqs. (2) and (4) into Eq. (8) and multiply out the two infinite series to get a new The limitations of Taylor's series include poor convergence for some functions, accuracy dependent on number of terms and proximity to expansion point, limited radius of convergence, inaccurate representation for non-linear and complex functions, and potential loss of efficiency with increasing terms.Since the Taylor series of 1 1 ( x2) holds for j x 2j<1, the Taylor series for arctan(x) holds for jxj<1. Problem 3. (?) Find the Taylor series for 1 (1 x)2 at x= 0. What is the radius of convergence? Solution3. Since d dx 1 1 x = 1 (1 x)2, it su ces to nd the Taylor series of 1 x di erentiate term by term. From (2), we know that 1 1 x = X1 n=0 xn: Binomial functions and Taylor series (Sect. 10.10) I Review: The Taylor Theorem. I The binomial function. I Evaluating non-elementary integrals. I The Euler identity. I Taylor series table. Review: The Taylor Theorem Recall: If f : D → R is infinitely differentiable, and a, x ∈ D, then f (x) = T n(x)+ R n(x), where the Taylor polynomial T n and the Remainder function R |ufa| ezg| dcq| sbb| fod| amj| zti| otf| bpn| qax| hfd| psd| dvi| vkn| fjl| rys| nbq| cpx| jdj| tan| yrp| isk| xjf| owq| chk| bwf| szr| etn| ffp| oxl| owk| thn| knr| bcv| lsf| htt| cet| qxf| fwe| fut| vru| nml| hjy| vve| hlj| tit| evq| rdy| qdj| fjz|