In this problem, you will investigate the error in the $n^{\rm th}$ degree Taylor approximation, $P_n(x)$, to $\ln\!\left(x+1\right)$ about $0$ for various values of $n$.

(a) Let $E_1 = \ln\!\left(x+1\right) - P_1(x) = \ln\!\left(x+1\right) - (x)$. Using a calculator or computer, graph $E_1$ for $-0.1 \leq x \leq 0.1$, and notice what shape the graph is. Then use the Error Bound for Taylor polynomials to find a formula for the maximum error, as a function of $x$, in this case:
$|E_1(x)| \le$ .

Graph the actual error $|E_1(x)|$ and your error bound together to see that the error is in fact below the maximum error bound (but close to it).

(b) Let $E_2=\ln\!\left(x+1\right) -P_2(x) = \ln\!\left(x+1\right) - (x-\frac{x^{2}}{2})$. Choose a suitable range and graph $E_2$ for $-0.1\leq x \leq 0.1$ . Again, notice what shape the graph of $E_2$ is. Then use the Error Bound for Taylor polynomials to find a formula for the maximum error, as a function of $x$, in this case:
$|E_2(x)| \le$ .

Graph the actual error $|E_2(x)|$ and your error bound together to see that the error is in fact below the maximum error bound (but close to it).

You can earn partial credit on this problem.