How to show that ${\lim_{n\to\infty}{e^{-t\sqrt{n}}\bigg({\frac{1}{1-\frac{t\sqrt{n}}{n}}\bigg)}^n}\large}=e^{\frac{t}{2}}$?

Question

How to show the following limit? $${\lim_{n\to\infty}{e^{-t\sqrt{n}}\bigg({\frac{1}{1-\frac{t\sqrt{n}}{n}}\bigg)}^n}\large}=e^{\frac{t}{2}}$$

I tried Taylor expansion and tried exponentiating. But I do not see how to go on.

I tried to solve this problem. Last step stuck. https://math.stackexchange.com/questions/2268847/moment-generating-function-of-sample-mean-and-limiting-distribution/2269257#2269257 — , Apr 08 '19 at 00:16
The expression $(..)^n$ resembles the expression which has limit $e^x$. See if you can manipulate it. — herb steinberg, Apr 08 '19 at 00:22
You should put the context in the body of of your question, along with an indication of your attempts. I'm afraid this is going to be closed otherwise. — saulspatz, Apr 08 '19 at 00:23
I think the whole thing is one rather than $e^{(\frac{t}{2})}$ — , Apr 08 '19 at 00:32
Unless I'm missing something, doesn't this only work if $t=0,1$? — R. Burton, Apr 08 '19 at 00:58

DINEDINE · Accepted Answer · 2019-04-08T01:11:44.780

2

$$\log\bigg(\big(1-\frac{t}{\sqrt{n}}\big)^{-n}\bigg)+\log{\exp(-t\sqrt{n }})=-n\log{\left(1-\frac{t}{\sqrt{n}}\right)}-t\sqrt{n}=t\sqrt{n}+\frac{t^2}{2}-t\sqrt{n}+o(1)=\frac{t^2}{2}+o(1)$$

I get $\exp{\left(\frac{t^2}{2}\right)}$ as limit.

edited Apr 08 '19 at 01:11

answered Apr 08 '19 at 00:47

DINEDINE

6,141

How to show that ${\lim_{n\to\infty}{e^{-t\sqrt{n}}\bigg({\frac{1}{1-\frac{t\sqrt{n}}{n}}\bigg)}^n}\large}=e^{\frac{t}{2}}$?

1 Answers1