History of the power series for $e^x$ and compound interest As discussed in How did Bernoulli approximate $e$?, Bernoulli showed that $2\frac{1}{2} < e < 3$ in this paper:
https://books.google.com/books?id=s4pw4GyHTRcC&pg=PA222#v=onepage&q&f=false
He gives a formula which we would write as:
$$
a + b + \frac{b^2}{2!\cdot a} + \frac{b^3}{3!\cdot a^2} + \frac{b^4}{4!\cdot a^3} + \frac{b^5}{5!\cdot a^4} + \ldots
$$
for the amount a creditor would receive for an investment of $a$ at annual interest $b$ for one year with interest compounded continually. Once you realise that the interest rate, $r$ say, is not $b$ but $\frac{b}{a}$ (i.e.,  $b$ is a sum of money not a ratio), this is what you'd expect:
$$
a (1 + \frac{r}{1!} + \frac{r^2}{2!} + \frac{r^3}{3!} + \ldots) =  a  e^{r}.
$$
My questions are: (1) am I right in thinking that Bernoulli is not claiming to prove this formula in this paper, but that he takes it as known? (2) how was the formula first proved? It is fairly easy to see that the amount at the end of the year should be given by the following limit (assuming it exists)
$$
\lim_{n \rightarrow \infty} \left(1 + \frac{r}{n}\right)^n
$$
but it is not particularly easy to prove algebraically that the coefficient of $r^n$ in this sequence of polynomials tends to $\frac{1}{n!}$. Was the equivalence of the two formulas first proved by investigating these coefficients or by modelling continuous compound interest using integration, or what?
 A: First, consider $a=1$ to make calculations easier.  Then consider $b,r=1$.  We would then get the taylor series expansion for $e$.
Keep in mind that Bernoulli did not know the exact value of $e$.  But with this series, he could approximate $e$.
I believe the series is derived as follows:

You start with $a$ amount of cash.  You earn a rate $\frac {100} r$% (rate should be less than one) amount of cash every $\frac{1year}{r}$ so that earning lower rates resulted in more frequent interest.
For example, if $r=1$, then you would earn $100$% interest after 1 year.
If $r=2$, then you would earn $50$% interest every 6 months, or twice in one year.
And for $r=2$, we see a mathematical series.  We start with $a$ dollars.  Then we earn $b$ dollars ($b=a*\frac 1r$, formula for compound interest) after the first 6 months.  Then we earn $\frac{b^2}{2!a}$, if I am not mistaken.
Our final amount of money earned would be$$S_2=a+b+\frac{b^2}{2!a}$$But Bernoulli noticed that we got the following series:$$S_{r}=a+b+\frac{b^2}{2!a}+...\frac{b^r}{r!a^{r-1}}$$
And taking this to $\infty$ gives us the Taylor Series of $ae^r$
He noted that the series approached a sort of limit, $2.5<e<3$, which he calculated via plugging in larger and larger $r$'s, if I'm correct.
This may not be the exact way Bernoulli solved this, but it is my best answer.
