Taylor Series

[Mooch22]

The Taylor series about x=0 for a certain function f converges to f(x) for all x in the interval of convergence. The nth derivative of f at x=0 is given by...

(f^n)(0) = (((-1)^(n+1))(n+1)!)/((5^n)((n-1)^2)) for n greater than or equal to 2.
Above stands for f to the n of zero.

The graph of f has a horizontal tangent line at x=0, and f(0)=6.

a.) Determine whether f has a relative maximum, a relative minimum, or neither at x=0. Justify your answer.

b.) Write the third-degree Taylor polynomial for f about x=0.

c.) Find the radius of convergence of the Taylor series for f about x=0. Must show work.

 

[skeeter]

a. determine the sign of f"(0) ... what does that tell you?

b. \(\displaystyle T_3(x) = f(0) + f'(0)x + \frac{f"(0)x^2}{2!} + \frac{f^3(0)x^3}{3!}\)

c. use the ratio test for convergence ...

\(\displaystyle lim_{n\rightarrow\infty}|\frac{a_{n+1}}{a_n}| < 1\), where

\(\displaystyle a_{n+1} = \frac{f^{n+1}(0)x^{n+1}}{(n+1)!}\)

and

\(\displaystyle a_n = \frac{f^n(0)x^n}{n!}\)