AP Calculus BC 10.14: Taylor Series Forms

Practice AP Calculus BC questions on deriving initial terms and general terms for Taylor or Maclaurin series from derivative information.

[Maximum number: 3]

A function f has derivatives of all orders for -1<x<1. The derivatives of f satisfy the conditions above. The Maclaurin series for f converges to f(x) for |x|<1.

Show that the first four nonzero terms of the Maclaurin series for f are $x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}$ , and write the general term of the Maclaurin series for f.

f(0)=0
$f^{\prime}(0)=1$ $f^{\prime \prime}(0)=-1(1)=-1$ $f^{\prime \prime \prime}(0)=-2(-1)=2$ $f^{(4)}(0)=-3(2)=-6$
The first four nonzero terms are
$0+1 x+\frac{-1}{2!} x^{2}+\frac{2}{3!} x^{3}+\frac{-6}{4!} x^{4}=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}$ .
The general term is $\frac{(-1)^{n+1} x^{n}}{n}$ .

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LIM-8.E—Represent a function as a Taylor series or a Maclaurin series question 1