Help Needed: Checking & Solving Power Series for f(x) = x*ln(1+x)

[tandoorichicken]

Two parts to this problem. On the first part I need someone to check my work, and I need help on solving the second part.

(a) Find a power series representation for f(x) = ln(1+x).
$$\frac{df}{dx} = \frac{1}{1+x} = \frac{1}{1-(-x)} = 1-x+x^2-x^3+x^4-...$$
$$\int_{n} \frac{1}{1+x} = \int_{n} (1-x+x^2-x^3+...)\dx = x-\frac{x^2}{2}+\frac{x^3}{3}-...+C = \sum^{\infty}_{n=0} \frac{(-x)^{n+1}}{n+1} +C$$
sub x=0 in original equation: ln(1+0) = ln(1) = 0 = C.
$$\ln(1+x) = \sum^{\infty}_{n=0} \frac{(-x)^{n+1}}{n+1}$$
with radius of convergence = 1.

(b) Find a power series representation for f(x) = x*ln(1+x).
If this function is differentiated, you get
$$\ln(1+x) + \frac{x}{x+1}$$
which is the same as a sum of power series
$$\sum^{\infty}_{n=0} \frac{(-x)^{n+1}}{n+1} + \sum^{\infty}_{n=0} (-x)^{n+1}$$
have I gone too far? or where do I go from here?
I know I will eventually have to integrate back to get the series for the original function.

 

[StatusX]

The first series looks right except for a sign, but why don't you just replace n+1 by n and run the index from 1 to infinity? For the second one, you can just multiply each term from the expansion for ln(1+x) by x.

 

[Hurkyl]

(a) looks fine (except for the sign), though it's usually good form to reindex the power series, and rearrange the terms, so that it looks like

$$\sum_{n = ?}^{\infty} (\mathrm{something}) x^n$$

(b) You don't have to do any differentiation at all... but you could do it that way if you really want to, by combining the two sums.