Editing Jacob Bernoulli (section)

==Discovery of the mathematical constant ''e''==
In 1683, Bernoulli discovered the constant {{mvar|[[e (mathematical constant)|e]]}} by studying a question about [[compound interest]] which required him to find the value of the following expression (which is in fact {{math|''e''}}):<ref>Jacob Bernoulli (1690) "Quæstiones nonnullæ de usuris, cum solutione problematis de sorte alearum, propositi in Ephem. Gall. A. 1685" (Some questions about interest, with a solution of a problem about games of chance, proposed in the ''Journal des Savants'' (''Ephemerides Eruditorum Gallicanæ''), in the year (anno) 1685.**), ''Acta eruditorum'', pp. 219–23.  [https://books.google.com/books?id=s4pw4GyHTRcC&pg=PA222 On p. 222], Bernoulli poses the question:  ''"Alterius naturæ hoc Problema est:  Quæritur, si creditor aliquis pecuniæ summam fænori exponat, ea lege, ut singulis momentis pars proportionalis usuræ annuæ sorti annumeretur; quantum ipsi finito anno debeatur?"'' (This is a problem of another kind:  The question is, if some lender were to invest [a] sum of money [at] interest, let it accumulate, so that [at] every moment [it] were to receive [a] proportional part of [its] annual interest; how much would he be owed [at the] end of [the] year?)  Bernoulli constructs a power series to calculate the answer, and then writes: ''" ... quæ nostra serie [mathematical expression for a geometric series] &c. major est.  ... si ''a''=''b'', debebitur plu quam {{sfrac|2|1|2}}''a''  & minus quam 3''a''."'' ( ... which our series [a geometric series] is larger [than]. ... if ''a''=''b'', [the lender] will be owed more than {{sfrac|2|1|2}}''a'' and less than 3''a''.)  If ''a''=''b'', the geometric series reduces to the series for ''a'' × ''e'', so 2.5 < ''e'' < 3.  (** The reference is to a problem which Jacob Bernoulli posed and which appears in the ''Journal des Sçavans'' of 1685 at the bottom of [http://gallica.bnf.fr/ark:/12148/bpt6k56536t/f307.image.langEN page 314.])</ref><ref>{{Cite web|url = http://www-history.mcs.st-and.ac.uk/HistTopics/e.html|title = The number e |author1=J J O'Connor |author2=E F Robertson |publisher = St Andrews University|access-date = 2 November 2016}}</ref>

:<math>\lim_{n\to\infty} \left( 1 + \frac{1}{n} \right)^n</math>

One example is an account that starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5<sup>2</sup>&nbsp;=&nbsp;$2.25. Compounding quarterly yields $1.00×1.25<sup>4</sup>&nbsp;=&nbsp;$2.4414..., and compounding monthly yields $1.00×(1.0833...)<sup>12</sup>&nbsp;=&nbsp;$2.613035....

Bernoulli noticed that this sequence approaches a limit (the [[Compound interest#Force of interest|force of interest]]) for more and smaller compounding intervals. Compounding weekly yields $2.692597..., while compounding daily yields $2.714567..., just two cents more. Using {{math|''n''}} as the number of compounding intervals, with interest of 100% / {{math|''n''}} in each interval, the limit for large {{math|''n''}} is the number that [[Leonhard Euler|Euler]] later named {{math|''e''}}; with ''continuous'' compounding, the account value will reach $2.7182818.... More generally, an account that starts at $1, and yields (1+{{math|R}}) dollars at [[Interest#Compound interest|compound interest]], will yield {{math|''e''}}<sup>{{math|R}}</sup> dollars with continuous compounding.