I keep coming across these limits - the context is moment generating functions and the Central Limit Theorem, but I'm guessing it's a more general question - here is one example (from the proof for the weak law of large numbers):
$$(1 - \frac{t^2}{2n} + o(\frac{t^2}{2n}))^n \rightarrow e^\frac{-t^2}{2} \\ \\ as \\ {n \rightarrow \infty}$$
I have not been able to find instructions that explain a few details which is keeping me from feeling fully comfortable with it (and it's almost guaranteed to show up in some form on the test).
My questions are
- What exactly is the o term? I understand it's some kind of remainder term when doing a Taylor expansion, but what rules does it follow? I've seen examples where t is moved outside of the o term in calculations. I've also seen the following (where the the t term doesn't even appear in the o term)
$$(1 + \frac{t^2}{6n} + o(\frac{1}{n}))^n \rightarrow e^\frac{t^2}{6} \\ \\ as \\ {n \rightarrow \infty}$$
Can the Taylor expansion be written out as an arbitrary number of terms as long as you add the appropriate o term? That's the impression I'm getting (i.e. I could continue up to t^3, t^4 etc I just have to modify the o term).
- If the n was not in the denominator, would it still converge. i.e.
$$ ?? (1 - \frac{t^2}{2} + o(\frac{t^2}{2}))^n \rightarrow e^\frac{-t^2}{2} ?? \\ \\ as \\ {n \rightarrow \infty}$$
- In the examples I've seen where this limit is used, it always only involves one term. I.e. either the first t term is removed (due to having a 0 product, as in the examples above), or only the first term is expanded like this:
$$(1 + \frac{ta}{n} + o(\frac{t}{n}))^n \rightarrow e^{ta} \\ \\ as \\ {n \rightarrow \infty}$$
What would happen if both the t and t^2 where included - would it still converge the same (or does it specifically need to be 1 + a single t term + the oterm)? i.e. something like:
$$??(1 + \frac{ta}{n} + \frac{(ta)^2}{2n} + o(\frac{t}{n}))^n \rightarrow e^{ta}?? \\ \\ as \\ {n \rightarrow \infty}$$
I appreciate any help, I just can't seem to figure these details out and they are throwing a wrench in my apparatus...thanks!!