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1 year ago in Calculus , Mathematical Analysis By Sourabh

What is the name of this transformed series f(x) = c0 + c1 x + c2 x^2 + c3 x^3 + …?

I'm revisiting foundational material in analysis. This series form appears ubiquitously, but I hear it called by different names power series, Taylor series, Maclaurin series. What is the most general, correct term for *f(x) = c0 + c1x + c2x^2 + ...*, and how do the more specific names relate?

 

SeriesExpansion PowerSeries TaylorSeries

All Answers (1 Answers In All)

By Rinku Answered 1 year ago

The most general term is a power series. Think of it as the overarching category. The names Taylor and Maclaurin specify how you determine the coefficients. If you calculate the coefficients c_n using derivatives of the function at a specific point *a* (i.e., c_n = f^(n)(a)/n!), it becomes a Taylor series about x = a. The Maclaurin series is just the special case where *a = 0*. So in your notation, it's a power series; it becomes a Taylor/Maclaurin series only when those coefficients have that specific derivative-based relationship to a function.

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