My Math Forum Effective interest rate compounded every two months

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July 3rd, 2011, 06:55 PM   #11
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Re: Effective interest rate compounded every two months

Quote:
 Originally Posted by Erimess I'll have to trust how you got there. We were just shown a chart that made m a 10, then 100, then 1000, etc., and showed how it tended towards e. But I've never learned why e is what it is, only how to use it for this one thing.
It was never explained satisfactorily to me until college! Suffice it to say that it's a very useful number, and that it is equal to 1/0! + 1/1! + 1/2! + 1/3! + ... = 1 + 1 + 1/2 + 1/6 + ....

July 3rd, 2011, 08:09 PM   #12
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Re: Effective interest rate compounded every two months

Quote:
 Originally Posted by Erimess I'll have to trust how you got there...
One way is as follows:

$\lim_{m\to\infty}$$\(1+\frac{r}{m}$$^m\)=L$

$e^{\ln$$\lim_{m\to\infty}\(\(1+\frac{r}{m}$$^m\)\) }=L$

$e^{$$\lim_{m\to\infty}\(\ln\(\(1+\frac{r}{m}$$^m\) \)\)}=L$

$e^{$$\lim_{m\to\infty}\(m\ln\(1+\frac{r}{m}$$\)\)}=L$

$e^{$$\lim_{m\to\infty}\(\frac{\ln\(1+\frac{r}{m}$$ }{\frac{1}{m}}\)\)}=L$

Application of L'Hôpital's rule gives:

$e^{$$\lim_{m\to\infty}\(\frac{-\frac{r}{m^2+mr}}{-\frac{1}{m^2}}$$\)}=L$

$e^{$$\lim_{m\to\infty}\(\frac{m^2r}{m^2+mr}$$\)}=L$

$e^{$$\lim_{m\to\infty}\(\frac{r}{1+\frac{r}{m}}$$\ )}=L$

$e^r=L$

 July 3rd, 2011, 10:00 PM #13 Senior Member   Joined: Apr 2011 From: USA Posts: 782 Thanks: 1 Re: Effective interest rate compounded every two months I don't know what you're doing. I know the infinity symbol, but I don't know how to use e and I don't know what ln means. (I can use those to solve for specific stuff that I have memorized, and I do mean just memorized plug n chug with no knowledge behind it.) So you lost me from the start. You don't need to try to explain it - you'd have to explain from scratch too many things I don't know. Since I've forgotten most of high school, it's easier to just pretend I stopped after 9th grade algebra. (Yes, I know more than that, but it's more specialized business stuff.)

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