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By Russell L. Herman

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 easy Harmonic OscillatorSecond Order Linear Differential EquationsLRC CircuitsDamped OscillationsForced SystemsCauchy-Euler EquationsNumerical recommendations of ODEsNumerical Read more...

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

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Two series which occur often are the geometric series and the binomial series. We will discuss these next. A geometric series is of the form ∞ ∑ arn = a + ar + ar2 + . . + arn + . . 87) n =0 Here a is the first term and r is called the ratio. It is called the ratio because the ratio of two consecutive terms in the sum is r. 24. For example, 1+ 1 1 1 + + +... 2 4 8 is an example of a geometric series. We can write this using summation notation, 1+ ∞ 1 1 1 + + +... = ∑ 1 2 4 8 n =0 1 2 n . Thus, a = 1 is the first term and r = 12 is the common ratio of successive terms.

Thus, 1 | x | = | x | < 1. n→∞ 1 We have obtained the same result as when we used the nth Root Test. 35. Find the radius of convergence of the series ∑∞ . n =1 n In this example, we have an expansion about x = 2. Using the nth Root Test we find that n n 3 ρ = lim | x − 2| = 3| x − 2| < 1. n→∞ n Solving for | x − 2| in this inequality, we find | x − 2| < 13 . Thus, the radius of convergence is R = 1 3 and the interval of convergence is 2 − 13 , 2 + As for the endpoints, we first test the point x = 3n ( 13 )n ∑∞ n =1 n 7 3.

8 we show the sum of the first 21 terms over the entire interval [−1, 1]. Note that there are problems with approximations near the endpoints of the interval, x = ±1. Such polynomial approximations are called Taylor polynomials. Thus, T3 ( x ) = 1 + x + x2 + x3 is the third-order Taylor polynomial approximation of f ( x ) = 1−1 x . With this example we have seen how useful a series representation might be for a given function. 7]. 8: Comparison of n ∑20 n=0 x for x ∈ [−1, 1]. 0 1 1− x (solid) to 28 mathematical methods for physicists Taylor series expansion.

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