Please check my proof (L'hôpital's rule)

[QuarkCharmer]

I should have made the title say "using L'h..." rather than what it is, I apologize.

Homework Statement

Prove the compounding interest formula from the following:
$$A=A_{0}(a+\frac{r}{n})^{nt}$$

Homework Equations

The Attempt at a Solution

$$A=A_{0}(a+\frac{r}{n})^{nt}$$

$$lim_{n\to\infty}A_{0}(a+\frac{r}{n})^{nt}$$

$$A_{0}e^{t lim_{n\to\infty}n ln(a+\frac{r}{n})}$$

let u = $lim_{n\to\infty}n ln(a+\frac{r}{n})$

So now the desired function is $A = A_{0}e^{tu}$

$$u = lim_{n\to\infty} \frac{ln(1+\frac{r}{n})}{\frac{1}{n}}$$

Using L'hôpital's rule gives that:

$$u = lim_{n\to\infty} \frac{\frac{d}{dn}ln(1+\frac{r}{n})}{\frac{d}{dn} \frac{1}{n}}$$

$$u = lim_{n\to\infty}\frac{rn^{2}}{n^{2}+rn}$$

$$u = lim_{n\to\infty}\frac{\frac{rn^{2}}{n^{2}}}{\frac{n^{2}}{n^{2}}+\frac{rn}{n^{2}}}$$

$$u = lim_{n\to\infty} \frac{r}{1+\frac{r}{n}} = \frac{r}{1} = r$$

$$u = r$$

Now, re-substituting r back in for u yields:

$$A = A_{0}e^{rt}$$Which is the typical $Pe^{rt}$ formula we all know and love?

I don't want to look it up and spoil it because I think I am right. Not sure if you can raise e to the ln of something, but I don't see why not. It's not like it changes anything and I figure you have to get an e in there somehow.

 

[micromass]

Some remarks:

$$u = lim_{n\to\infty} \frac{ln(1+\frac{r}{n})}{\frac{1}{n}}$$

Why did you drop the a?? You have exchanged a by 1. Eventually it won't matter, but I don't see how you can drop it just like that.

Using L'hôpital's rule gives that:

$$u = lim_{n\to\infty} \frac{\frac{d}{dn}ln(1+\frac{r}{n})}{\frac{d}{dn} \frac{1}{n}}$$

$$u = lim_{n\to\infty}\frac{rn^{2}}{n^{2}+rn}$$

This is correct, but I don't see where you get it from.

Not sure if you can raise e to the ln of something, but I don't see why not. It's not like it changes anything and I figure you have to get an e in there somehow.

Yes, you can do it, and it's one of the great tricks in working with limits. Did you discover that trick all by yourself?? Great!

 

[vela]

I should have made the title say "using L'h..." rather than what it is, I apologize.

Homework Statement

Prove the compounding interest formula from the following:
$$A=A_{0}(a+\frac{r}{n})^{nt}$$

Homework Equations

The Attempt at a Solution

$$A=A_{0}(a+\frac{r}{n})^{nt}$$

$$lim_{n\to\infty}A_{0}(a+\frac{r}{n})^{nt}$$

$$A_{0}e^{t lim_{n\to\infty}n ln(a+\frac{r}{n})}$$

let u = $lim_{n\to\infty}n ln(a+\frac{r}{n})$

So now the desired function is $A = A_{0}e^{tu}$

$$u = lim_{n\to\infty} \frac{ln(1+\frac{r}{n})}{\frac{1}{n}}$$

Why did you set a=1?

 

[QuarkCharmer]

Oh wow, yeah that "a" should have been a "1" from the start. I typed out the formula I was starting with incorrectly and then copy/paste the errors further into the problem until I finally typed a line by hand and fixed it!

I'm a horrible latex-er?

This is correct, but I don't see where you get it from.

Truth be told, I didn't want to type out the intermediate steps where I took the derivative and simplified. I know that step to be correct and this is not a real "proof" that I am turning into anyone. I am just doing some extra problems from Stuart 6E 7.8 (This one is #85).

Aside from the missing intermediate steps, and the horrid typo (I swear it's correct on my handwritten paper), everything else is okay? More specifically, if I were to document all the steps in a proof based writing (which I have no idea how to do), it would make sense how I came to the conclusion?

Thanks for all the input.

 

[micromass]

Aside from the missing intermediate steps, and the horrid typo (I swear it's correct on my handwritten paper), everything else is okay? More specifically, if I were to document all the steps in a proof based writing (which I have no idea how to do), it would make sense how I came to the conclusion?

Yes, everything is clear and correct!

 

[QuarkCharmer]

Yes, you can do it, and it's one of the great tricks in working with limits. Did you discover that trick all by yourself?? Great!

I thought so but I doubt it. I think I remember a similar tactic used in a proof of something on khanacademy and I just stole the idea.

Thanks for the help!