[SOLVED] Big O for functions of epsilon

• Feb 3rd 2008, 11:18 PM
acseng
[SOLVED] Big O for functions of epsilon
I'm needing a direction for this problem:

show terms correct to O $(\epsilon^2)$ for $\frac{1+\epsilon}{1-\epsilon}$ , $(1+\epsilon+\epsilon^2)^{1/2}$ , and $sin(1+\epsilon)$.

I'm thinking a series expansion for each expression. But I'm just a little confused. Any help or suggestions would be greatly appreciated.

Thanks!
• Feb 4th 2008, 03:22 AM
mr fantastic
Quote:

Originally Posted by acseng
I'm needing a direction for this problem:

show terms correct to O $(\epsilon^2)$ for $\frac{1+\epsilon}{1-\epsilon}$ , $(1+\epsilon+\epsilon^2)^{1/2}$ , and $sin(1+\epsilon)$.

I'm thinking a series expansion for each expression. But I'm just a little confused. Any help or suggestions would be greatly appreciated.

Thanks!

You need a series for each - up to the $\epsilon^2$ term.

For the first, I'd first re-write it as $\frac{2}{1 - \epsilon} - 1$.

For the second, I'd first re-write it as $\left( \left[ \epsilon + \frac{1}{2} \right]^2 + \frac{3}{4} \right)^{1/2}$.

For the first, after the re-write I'd use the formula for an infinite geometric series.

For the second (after the re-write) and third, I'd first get the standard expansions for $(x^2 + a)^{1/2}$ and $\sin x$ and then make the appropriate substitutions ......