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Chapter6 Power SeriesPower Series are one of the most useful type of Series in analysis. For example,we can use them to define transcendental functions such as the exponential andtrigonometric functions (and many other less familiar functions). IntroductionA Power Series (centered at 0) is a Series of the form n=0anxn=a0+a1x+a2x2+ +anxn+..where theanare some coefficients. If all but finitely many of theanare zero,then the Power Series is a polynomial function, but if infinitely many of theanarenonzero, then we need to consider the convergence of the Power basic facts are these: Every Power Series has a radius of convergence 0 R , which depends on the coefficientsan. The Power Series converges absolutelyin|x|< Rand diverges in|x|> R, and the convergence is uniform on every interval|x|< where 0 < R. IfR >0, the sum of the Power Series is infinitelydifferentiable in|x|< R, and its derivatives are given by differentiating the originalpower Series Series work just as well for complex numbers as real numbers, and arein fact best viewed from that perspective, but we restrict our attention here toreal-valued Power nition (an) n=0be a sequence of real numbers andc R.

The power series in Definition 6.1 is a formal expression, since we have not said anything about its convergence. By changing variables x→ ( x−c ), we can assume

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