By Russell L. Herman

Advent and ReviewWhat Do i have to be aware of From Calculus?What i would like From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe basic Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical suggestions of ODEsNumericalRead more...

summary: creation and ReviewWhat Do i have to understand From Calculus?What i want From My Intro Physics Class?Technology and TablesAppendix: Dimensional AnalysisProblemsFree Fall and Harmonic OscillatorsFree FallFirst Order Differential EquationsThe basic Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical ideas of ODEsNumerical ApplicationsLinear SystemsProblemsLinear AlgebraFinite Dimensional Vector SpacesLinear TransformationsEigenvalue ProblemsMatrix formula of Planar SystemsApplicationsAppendix: Diagonali

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Instead of explicitly computing the Taylor series expansion for this function, we can make use of an already known function. We first write f ( x ) as a function of x − 1, as we are expanding about x = 1; that is, we are seeking a series whose terms are powers of x − 1. 1 1 This expansion is easily done by noting that x+ 2 = ( x −1)+3 . Factoring out a 3, we can rewrite this expression as a sum of a geometric series. Namely, we use the expansion for g(z) = = 1 1+z 1 − z + z2 − z3 + . . 100) 31 32 mathematical methods for physicists and then we rewrite f ( x ) as f (x) = = = = 1 x+2 1 ( x − 1) + 3 1 1 3[1 + 3 ( x − 1)] 1 1 .

Interval and radius of convergence. The number R is called the radius of convergence of the power series and ( a − R, a + R) is called the interval of convergence. Convergence at the endpoints of this interval must be tested for each power series. In order to determine the interval of convergence, one needs only to note that when a power series converges, it does so absolutely. So, we need only ∞ n n test the convergence of ∑∞ n=0 | cn ( x − a ) | = ∑n=0 | cn || x − a | . This is easily done using either the ratio test or the nth root test.

N=0 cn ( x − a ) converges for | x − a | < R and diverges for | x − a| > R. Interval and radius of convergence. The number R is called the radius of convergence of the power series and ( a − R, a + R) is called the interval of convergence. Convergence at the endpoints of this interval must be tested for each power series. In order to determine the interval of convergence, one needs only to note that when a power series converges, it does so absolutely. So, we need only ∞ n n test the convergence of ∑∞ n=0 | cn ( x − a ) | = ∑n=0 | cn || x − a | .