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Taylor Series and Polynomials : Removable Singularity, Continuity and Finding Terms

Consider the function f(x)=x/(ex-1)+x/2.

(a) f has a so-called "removeable singularity" at x=0, where it is (so far) undefined. What value should we assign to f(0) to make f continuous at x=0?

(b) With this taken care of, f actually has a Taylor series about x=0. Find the first 10 terms or so of this Taylor series (use CALCULATOR/MAPLE)

(c) What pattern do you notice in the degrees of the terms in the Taylor polynomials?

(d) Prove the property that you noticed in (c).

Solution Summary

Taylor Series and Polynomials, Removable Singularities, Continuity and Finding Terms are investigated. The solution is detailed and well presented.