# Taylor expansion in infinity?

1. Nov 16, 2014

### Vrbic

1. The problem statement, all variables and given/known data
How to use Taylor series for condition x>>1? For example $f(x)=x\sqrt{1+x^2}(2x^2/3-1)+\ln{(x+\sqrt{1+x^2})}$

2. Relevant equations

3. The attempt at a solution
I try to derived it and limit to infinity...for example first term $\frac{x^4}{3\sqrt{1+x^2}}$. Limit this to infinity is obviously infinity. Any advice?
Thank you.

2. Nov 16, 2014

### Staff: Mentor

Well, the function goes to infinity, so every approximation should do the same.
What exactly do you want to calculate or approximate where?

In general, to get good approximations for large x, you can calculate the taylor series of f(1/x) and evaluate it around 0. That won't work with divergent series, however, where you might need a Laurent series.

3. Nov 16, 2014

### Vrbic

Nice trick, I tried it but for all derivative limit to 0 is divergent...for example 1.derivative of my f(1/x) is $-\frac{1+1/x^2}{\pi x^4}$. Yes I would like to examine a behaviour of this function for large x and approximate it for this case.

4. Nov 16, 2014

### Staff: Mentor

Well you'll need a Laurent series here, as I mentioned. If it stops, then you have a good approximation.