# Taylor expanding equation

1. Apr 27, 2016

### spacetimedude

1. The problem statement, all variables and given/known data
Consider the position vector of a mass m at height h above the Earth's surface to be $\underline{r}=(R+h)\underline{e}_z$ where R is the radius of the Earth.
Make a Tylor expansion in h/R <<1 of the gravitational field
$$\underline{G}(\underline{r})=\frac{-GM\underline{r}}{r^3}$$
to show that the gravitational force on a mass m can be written as $$\underline{F} \approx mg\underline{e}_z$$ giving an expression for g in terms of R and M, the mass of the Earth.
Find the first correction (h-dependent) to the gravitational force.

2. Relevant equations

3. The attempt at a solution
I am having trouble understanding how to expand the function in h/R<<1.

We can rewrite $$\underline{G}(\underline{r})=\frac{-GM(R+h)}{(R+h)^3}\underline{e}_z$$ which then can then be expressed as $$\underline{G}(\underline{r})=\frac{-GM}{R^2}(\frac{h}{R}+1)^{-2}\underline{e}_z$$
I am not sure how to proceed with the expansion.
Any explanation will be appreciated.

2. Apr 27, 2016

### TSny

Let $x = h/R$. You need to make a Taylor series expansion of the function $f(x) = (x+1)^{-2}$.