I know that the limit as n goes to infinity of ((n-2)/n)^n=e^-2 but I do not know the steps to arrive there. Can someone help me?
Also how do you enter math so that it actually looks like math and not the "gobbelty gook" above
I know that the limit as n goes to infinity of ((n-2)/n)^n=e^-2 but I do not know the steps to arrive there. Can someone help me?
Also how do you enter math so that it actually looks like math and not the "gobbelty gook" above
Do you already know that $\displaystyle \lim_{n\to\infty}\left(1+\frac{1}{f(n)}\right)^{f( n)}=e $ for any function $\displaystyle f(n)\,\,\,s.t.\,\,\, f(n)\xrightarrow [n\to\infty]{} \pm\infty$ ? If you do then:
$\displaystyle \left(\frac{n-2}{n}\right)^n=\left[\left(1+\frac{1}{-n\slash 2}\right)^{-n\slash 2}\right]^{-2}\xrightarrow [n\to\infty]{}e^{-2}$
Tonio
To answer your second question, use "LaTex" by staring with "[math ]" and ending with "[/math ]" without the last space.
For example, $\displaystyle \lim_{n\to\infty}\left(\frac{n- 2}{n}\right)^n$.
Click on that to see the code. There is also a "LaTex tutorial" on this forum.
To answer your first question, rewrite the formula as $\displaystyle \left(1- \frac{2}{n}\right)^n$ and let $\displaystyle m= \frac{n}{2}$ so n= 2m. Now it becomes $\displaystyle \left(1- \frac{1}{m}\right)^{2m}$$\displaystyle = \left[\left(1- \frac{1}{m}\right)^m\right]^2$.
Before continuing, I would need to know- what definition of "e" are you using?
Once again, Tonio got in just ahead of me!
where e= the limit of (1+1/n) as n goes to infinity.
Thanks Tonio but it seems like you went through a few steps which I did not follow. Can someone fill in the gaps. I'm in the second calculus in college (series and multivariable calculus) and I just recently learned that that was the definition of e. I was never taught that so now I'm lost when it comes to these kind of limits.
Do you guys get paid for this. This seems a little too convenient. I was recently online and I tried to get tutored for math online and the fee was outrageous (something like $30 a month) and then I recently stumbled on this website. I guess what I'm trying to say is do you guys do this out of the goodness in your heart or do you get compensated?