Understanding little-o notation

[zonk]

I am currently reading Apostol volume 1, and I'm having trouble understanding little-o notation. I have searched the web for free resources, but can't find any. I don't quite understand his proof for theorem 7.8.e, and I don't understand his examples.

For instance, why does $\frac{1}{1 + g(x)} = 1 - g(x) + g(x)\frac{g(x)}{1 + g(x)} = 1+ g(x) + o(g(x))$ as g(x) approaches 0?

Also for his example, why does $o(-\frac{1}{2}x^2 + o(x^3)) = o(x^2)$?

Anyone know of any free resource that explains this lucidly?

 

[zonk]

I think I understand it better. -1/2x^2 is of order o(x^2), and since we are considering smaller orders o(x^2) + o(x^3) = o(x^2). Also the third term approaches zero and 1 - g(x) is of order o(g(x)). Is this understanding correct?

 

[zonk]

No, I still don't understand little-o. I tried some exercises and failed miseraby. Is it something we need to understand a future section in Apostol, or is it something books on real analysis present and can safely be skipped with respect to understanding the rest of Apostol Calculus volume 1 and 2?

 

[chiro]

No, I still don't understand little-o. I tried some exercises and failed miseraby. Is it something we need to understand a future section in Apostol, or is it something books on real analysis present and can safely be skipped with respect to understanding the rest of Apostol Calculus volume 1 and 2?

Just to confirm, is your little o(h(x)) defined to be lim (x->0) h(x)/x -> zero? If not what is it?

 

[zonk]

Yeah, except x can approach any limit, not just 0 or infinity. So f(x) = o(g(x)) means that f(x)/g(x) -> 0.

 

[yifli]

I am currently reading Apostol volume 1, and I'm having trouble understanding little-o notation. I have searched the web for free resources, but can't find any. I don't quite understand his proof for theorem 7.8.e, and I don't understand his examples.

For instance, why does $\frac{1}{1 + g(x)} = 1 - g(x) + g(x)\frac{g(x)}{1 + g(x)} = 1+ g(x) + o(g(x))$ as g(x) approaches 0?

Also for his example, why does $o(-\frac{1}{2}x^2 + o(x^3)) = o(x^2)$?

Anyone know of any free resource that explains this lucidly?

You can find the rigorous definition of little-o in Section 5 of Chapter 3 from Advanced Calculus by Loomis http://www.math.harvard.edu/~shlomo/

A function $f \in \mathbb{o}$ if $f(0)=0$ and $\left\|f(\alpha)\right\|/\left\| \alpha \right\| \rightarrow 0$ as $\alpha \rightarrow 0$

All functions in little-oh constitute a vector space