Study guide for Taylor series


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Below is a list of skills I hope you, the student, have acquired
from the chapter on taylor polynomials.


Proficiency at computing Taylor polynomials
  • from the definition
  • substitution (e.g. Taylor pol. of 1/(1+x2))
  • by combining known Taylor polynomials (adding/subtracting/multiplying them)
  • differentiating/integrating known Taylor polynomials
Manipulating little oh as in problems 60 & 63
Find a recurrence relation for the coefficients of the T.polynomial of a rational function, as in the example of the Fibonacci numbers. See problems 61 & 62
Convergence of Taylor series: in our class this means you must show that the remainder term Rnf(x) goes to zero as n goes to infinity.
  • Using Lagrange's formula: this is only practical for ex, sin(x), cos(x), sinh(x) and cosh(x) (see problem 67).
  • Substitution in a series which is known to converge. E.g. the series for sin(x2). (see problem 69)
  • Using the formula for the Geometric sum: the student should know how prove convergence of the geometric series, and minor variations, such as 1/(1-2x), 1/(1+x2) (by substituting -2x=t or x2=t).