In this post we will recall the readers the proof of the well-known formula of compound interest and how it is related with the exponential function. It is worth to mention that the famous mathematical constant e, was "discover" by **Jacob Bernoulli** in 1680, while studying the compound interest. We will show here this analysis.

The formula for the accumulated capital after n years of compounding an initial capital or principal x<sub>0</sub> with an  fixed anual interest  rate i, and with a compound frequency k, is
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https://steemitimages.com/DQmRuoyssQjVD5Bomqnns1LiCs4L6ptUWu84VUJPN2UvYMt/int1.png
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Now we show, how to prove this formula.

Let us suppose that you have an initial capital x<sub>0 </sub>and you deposit it with a fix interest rate i for a period of time, say one year. After one year you have x<sub>1</sub> amount of capital, where x<sub>1</sub>=x<sub>0</sub> (1+i). Now you reinvest x<sub>1</sub> for the same period of time and at the same interest. After the second year the accumulated capital is x<sub>2</sub>:
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x<sub>2</sub>=x<sub>1</sub>(1+i)=x<sub>0</sub>(1+i)<sup>2</sup>.
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So after n years, if the principal plus interest are compounded at the same fixed interest rate, the accumulated capital is
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x<sub>n</sub>=x<sub>n-1</sub>(1+i)= ... =x<sub>0</sub>(1+i)<sup>n</sup>.
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Now we consider a variation of the investment strategy. Your initial capital is x<sub>0</sub> and  you deposit it for a year with a fix interest rate i, however your capital and interests are paid after six months and you reinvest it. So after six month your accumulated capital is
<center>
https://steemitimages.com/DQmXMpo7R63L9duVCpgmvo7kXb9WuRchWDcJT3pYSpS8jP7/int2.png
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If you reinvest this amount (compounding the principal  and interest), in the next six months (one year after your initial deposit). The accumulated capital is
<center>
https://steemitimages.com/DQmegESMTASCf5okwi6scvEEdazuB8Ari13D3MKy7VYYfSM/int3.png
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If you keep this strategy after n years the accumulated capital is
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https://steemitimages.com/DQmcGen21JHgZJbk1hyE48VAsRfUCu5VUfTkR9s1MXCvpfi/int4.png
</center>
If you modified the strategy, so that the interests are received every month and and they are reinvested (compounded), then the accumulated capital  after one year is 
<center>
https://steemitimages.com/DQmbHWPpPVJEP6sQvUqS6nCT6cfaGA3Gui4mJGFUj6PpvQ2/int5.png
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and after n years, with the same strategy the accumulated capital is
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https://steemitimages.com/DQmYdgnAC4gu5X2Q2ZQiNd7rfwADq7Zj7hjVm4UtSCuLSqJ/int6.png
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Now suppose that the interests are paid daily and compounded after payment. The capital accumulated after n years is
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https://steemitimages.com/DQmc9W8ASAMU9oQkg8xmNBv9Mh3efnWNSixybDLRTTPzk5s/int7.png
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So what happens if the interests are paid every second and compounded, or if this is done continuously. What is the formula? The answer is given by the limit
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https://steemitimages.com/DQmRgKZRRAoTzXFEZ7wGwu5ASMvEc49KE6KwiYSvVJvi6uD/int8.png
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The following is a well-known limit:
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https://steemitimages.com/DQmWcvishb243XsA4Qn2NM395YFzXmf5ajszdk7r8wRWsqt/int9.png
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Where e is the famous constant, whose rational approximation is   2.71828... 
This is the limit, that the Swiss mathematician **Jacob Bernoulli** worked in 1680.

From this expression follows:
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https://steemitimages.com/DQmbG5uARswqyd8coDnFKLp85JLPgX494EN4MxPr6W4ZDfn/int10.png
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Hence x<sub>n</sub>=x<sub>0</sub>e<sup>in</sup>.

Using this expression, we can answer how long does it take to duplicate the initial capital, assuming we use continuous compounding. In fact, we need x<sub>n</sub>=2x<sub>0</sub>, then
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https://steemitimages.com/DQmVkuTKXCu3Mp8pix8GrUUe98aTMb2G8u2UcBym6wfcus9/int11.png
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Therefore
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https://steemitimages.com/DQmeQxu4mhZYMjynmcDrLn9LbLsx1KGuMf73xrzCpToY7pL/int12.png
</center>

References:
https://en.wikipedia.org/wiki/Compound_interest
[https://en.wikipedia.org/wiki/E_(mathematical_constant)](https://en.wikipedia.org/wiki/E_(mathematical_constant))

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All the formulae of this post were typed by myself in LaTeX.</h3>
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