# Where does this approximation come from?

1. Apr 11, 2013

### fereopk

$$\frac{\sqrt{1-a}}{\sqrt{1-b}}\approx \left ( 1-\frac{1}{2}a\right )\left ( 1+\frac{1}{2}b\right )$$

I know that the binomial approximation is first used,

$$\frac{\sqrt{1-a}}{\sqrt{1-b}}\approx \frac{1-\frac{1}{2}a}{1-\frac{1}{2}b}$$

But how does one approximate:

$$\frac{1-\frac{1}{2}a}{1-\frac{1}{2}b}\approx \left ( 1-\frac{1}{2}a\right )\left ( 1+\frac{1}{2}b\right )$$?

2. Apr 11, 2013

### slider142

Do you know the series for $\frac{1}{1-x}$ ?

3. Apr 11, 2013

### fereopk

No, unfortunately. Is there a name for this approximation?

4. Apr 11, 2013

### Curious3141

The expression can be rearranged to: $\displaystyle {(1-a)}^{\frac{1}{2}}{(1-b)}^{-\frac{1}{2}}$. Now apply the binomial approximation to each term.

5. Apr 11, 2013

### fereopk

Ahh, I see. Thanks!