- #1

sooyong94

- 173

- 2

## Homework Statement

Given that ##f(x)=(1+x) ln (1+x)##.

(a) Find the fifth derivative of f(x),

(b) Hence, show that the series expansion of f(x) is given by

##x+\frac{x^{2}}{2} -\frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20}##

(c) Find, in terms of r, an expression for the rth term, (r>=2) of the Maclaurin expansion for f(x).

## Homework Equations

Product rule, Maclaurin series

## The Attempt at a Solution

For (a) I have used product rule and simplified the answer as ##-6(1+x)^{-4}##

For part (b), I just have to plug in 0 into f(x) and up to the fifth derivative, right?

Part (c)... Now I'm stuck. I know the sign alternates each other, so I have to use the term

##(-1)^{r}## for that. The trouble is dealing with the denominator. It looks like a series, though the denominator looks like a series, but it doesn't look like an arithmetic nor geometric series. :/