Section Summary

In this section we have used known formulas for geometric sums to derive Taylor polynomials and Taylor series for the important functions `ln text[(] 1+x text[)]` and `text[arctan(] x text[)]`. Along the way we have made or confirmed the following discoveries:

• If `P_n text[(] x text[)]` is a Taylor polynomial for `f text[(] x text[)]`, then we can substitute a power function `x = p text[(] t text[)]` in both formulas, and `P_n(p text[(] t text[)])` will be a Taylor polynomial for `f(ptext[(] t text[)])`. [The degree of this new Taylor polynomial will not be `n` unless `p text[(] t text[)]` is linear.]

• If `P_n text[(] t text[)]` is a Taylor polynomial of degree `n` for `f text[(] t text[)]`, then

  ∫ 0 x P n ( t )   d t

  is a Taylor polynomial of degree `n+1` for

  ∫ 0 x f ( t )   d t .

• Substitution and integration can be used to find Taylor polynomials that would be very difficult to find by using the derivative formula for Taylor coefficients.

• Some Taylor series have limited intervals of convergence. Outside these intervals, the Taylor polynomials do not approximate the function from which they were derived, and therefore the corresponding Taylor series does not converge to that function.

• Taylor series found by substitution and integration inherit their intervals of convergence in a natural way. (Note that we have not yet settled all the questions that might arise about convergence at endpoints of these intervals.)