(a) Find a power series representation for
$f(x) = \ln(1 + x)$.
$$f(x) = \sum _{n=1}^{\infty } \frac{\left(-1\right)^{n-1}x^n}{n}$$
What is the radius of convergence, $R$?
$R = 1$
(b) Use part (a) to find a power series for
$f(x) = x \ln(1 + x)$.
$$f(x)=\sum _{n=2}^{\infty } \frac{\left(-1\right)^n x^n}{n-1}$$
What is the radius of convergence, $R$?
$R = 1$
(c) Use part (a) to find a power series for
$f(x) = \ln(x^2 + 1).$
$f(x)=\sum _{n=1}^{\infty}$ ?? <– This is the answer I'm having trouble finding out.
What is the radius of convergence, $R$?
$R = 1$
Best Answer
From (a), \begin{equation} f(x) = \ln (1 + x) = \sum_{n=1}^\infty \frac{\left(-1\right)^{n-1}\left(x^n\right)}{n}. \end{equation} Then, (c) becomes \begin{equation} \ln (x^2 + 1) = \ln(1 + x^2) = f(x^2) = \sum_{n=1}^\infty \frac{\left(-1\right)^{n-1}\left(x^{2n}\right)}{n}. \end{equation}