Chapter 9.10, Problem 82E

Investigation

(a) Find the power series centered at 0 for the function

$f(x)=\frac{\ln (x^2+1)}{x^2}$

(b) Use a graphing utility to graph f and the eighth-degree Taylor polynomial $P_8(x)$ for f.

(c) Use a graphing utility to complete the table, where

$F(x)={\displaystyle {\int }_0^x\frac{\ln (t^2+1)}{t^2}dt\text{and}}G(x)={\displaystyle {\int }_0^x{P}_8(t)dt}$.

x	0.25	0.50	0.75	1.00	1.50	2.00
F(x)
G(x)

(d) Describe the relationship between the graphs of f and $P_8$ and the results given in the table in part (c).