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Math Help - Proof for e^x

  1. #1
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    Proof for e^x

    I have a (possible) proof for that I'd like you guys to verify. I'm pretty sure my calculations are correct, but I'm not sure the premise that I said something like "now we just have to prove that lim... n (ln(1 + x/n)) = x". If my approach is completely incorrect, then could someone start me off in the right direction with a hint? Here's a direct link, so I don't burden viewers with slow load times. Thanks!
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  2. #2
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    Quote Originally Posted by Lord Voldemort View Post
    I have a (possible) proof for that I'd like you guys to verify. I'm pretty sure my calculations are correct, but I'm not sure the premise that I said something like "now we just have to prove that lim... n (ln(1 + x/n)) = x". If my approach is completely incorrect, then could someone start me off in the right direction with a hint? Here's a direct link, so I don't burden viewers with slow load times. Thanks!

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  3. #3
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    Quote Originally Posted by Lord Voldemort View Post
    I have a (possible) proof for that I'd like you guys to verify. I'm pretty sure my calculations are correct, but I'm not sure the premise that I said something like "now we just have to prove that lim... n (ln(1 + x/n)) = x". If my approach is completely incorrect, then could someone start me off in the right direction with a hint? Here's a direct link, so I don't burden viewers with slow load times. Thanks!

    There are many fuzzy things here:

    1) How can you use a^x=e^{x\ln a} if you haven't yet shown the premise and deduced properties of the exp. function? Not to mention all those derivatives and antiderivatives which usually come AFTER one knows what goes on with the exponential function and its inverse.

    2) \frac{d}{dx}\left(\ln \left(1+\frac{x}{n}\right)\right)'=\frac{1}{n}\fra  c{1}{1+\frac{x}{n}}=\frac{1}{n+x}. You wrote there a very strange x' (what does this mean? Is this 1 = the derivative of x?) and many other things.

    3) Among other things you pass to calculate limits and you write: \lim_{n\rightarrow \infty}\ln \left(1+\frac{x}{n}\right)=\ln 1 , which is justified IF you already know the inverse function of the exponential function AND if you already know it is continuous at 1. This is too much to assume, imo, since you still have to show your original premise.

    In fact e^x=\lim_{n\rightarrow \infty}\left(1+\frac{x}{n}\right)^n is pretty much the usual definition for many authors, BUT you can also try to use Newton's Binomial theorem with \left(1+\frac {x}{n}\right)^n and then pass to the limit: you'll get a power series for e^x...

    Tonio
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    I found a better solution, I think, but thanks for the post.
    I think the main problem was that I was assuming things that were based on the e^x function. #2 though, I was differentiating with respect to n, not x.
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  5. #5
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    Quote Originally Posted by Lord Voldemort View Post
    I found a better solution, I think, but thanks for the post.
    I think the main problem was that I was assuming things that were based on the e^x function. #2 though, I was differentiating with respect to n, not x.

    First, in your notes you always use \frac{d}{dx} , never \frac{d}{dn}.

    Second, you can't derivate wrt n: that's a discrete variable and you need a continuous one in order to be able to apply limits in the real numbers, as the definition of derivative requires.

    Tonio
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