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I'm trying to evaluate what compound interest approaches when the number of compound periods approaches infinity. That is, I'm interested in analytically evaluating Euler's number $e$:

$$\lim_{m \to \infty} \left(1 + \frac{1}{m}\right)^m$$

Do you have any suggestions or ideas regarding how I can approach evaluating this? Also, is there a general approach to solving this kind of limit?

nocomment
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    It could be the definition of the constant $e$ if you like. – GEdgar Sep 29 '20 at 21:42
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    What do you mean evaluate? Just plug in a large number, say $m=1~000~000$ to get an approximation for $e$. – K.defaoite Sep 29 '20 at 21:43
  • I updated the problem. What I'm interested in is how to analytically evaluate that the limit tends to number $e$. – nocomment Sep 29 '20 at 21:44
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    How to prove that the limit evaluates to $e$ will depend entirely on what your definition of $e$ is. How have you defined it? – Arthur Sep 29 '20 at 21:44
  • Number $e$ is defined as per usual, that is Euler's number. I guess what I'm really interested in is the analytical proof that this is the case, without making approximations by plugging in large values of $m$. – nocomment Sep 29 '20 at 21:48
  • @K.defaoite Thanks, I guess it does. Cheers – nocomment Sep 29 '20 at 21:49
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    "Number $e$ is defined as per usual, that is Euler's number." I can think of four or five distinct, usual definitions for that constant. And they all end up with the same name, so that's no help in narrowing it down. You need to be more specific. Is it defined as $\sum \frac1{n!}$? Is it defined via the solution to the differential equation $y'=y$? Is it defined as the base of the natural logarithm? Something else? Which is it? – Arthur Sep 29 '20 at 21:56
  • You know that the limit is $e$. So find the series for $e^x$ and then let $x=1$. – Steven Alexis Gregory Sep 29 '20 at 22:34

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