ccirclecopyrt[email protected]CONFIDENTIALccirclecopyrt[email protected]Week 10Learning Objective:•Power series (§11.8)•Representations of functions as power series (§11.9)•Taylor and Maclaurin series (§11.10)10.1Power seriesHaving been familiar with the basic concepts of convergence of sequences and series,we now pursue the idea of representing functions as sums of infinite series. In thiscourse, we focus on power series.Definition 10.1A series of the form∞summationdisplayn=0cn(x−a)n,(10.1)whereaandcnare real numbers, is called apower series in(x−a) or apowerseries centered ataor apower series abouta.Remark 10.1The symbolxin (10.1) denotes a variable.We want to find outthe values ofxsuch that the resulting series converge or diverge. When dealing withpower series, we make an agreement that 00is equal to 1. Then the power series (10.1)automatically converges whenx=aand converges toc0.To observe some extremes, let us examine the convergence of the power series∞∑n=0nnxnand∞∑n=0xnn!.117
118MA104Example 10.1For what values ofxis the power series∞∑n=0nnxnconvergent?
Example 10.2Determine the values ofxat which the series
Example 10.3Find all values ofxwhere the power seriesn(x+ 2)nconverges.