Approximatio ad Summam Terminorum Binomii (a + b)^n in Seriem expansi

Abraham de Moivre (1667–1754) circulated this seminal pamphlet privately in 1733 as a supplement to his Miscellanea Analytica. Abraham de Moivre addresses the long-standing problem of summing terms of the binomial expansion for large powers, a task essential to solving problems in games of chance. Moving beyond the exact limits for the sum of terms previously established by Jacob and Nicolaus Bernoulli, this work provides a practical and elegant method for approximation.

The investigation yields several foundational results in probability and mathematical analysis. It begins by deriving an expression for the ratio of the middle term of \((1+1)^n\) to the total sum \(2^n\), initially in terms of a converging series. Crucially, it incorporates the discovery (communicated by James Stirling) that the constant \(B\) in this approximation is equal to \( \sqrt{\pi/2} \), where \( \pi \) represents the circumference of a circle of unit diameter. This leads to the famous result that the middle term is approximately \( 2 / \sqrt{\pi n} \).

The paper then presents a general expression for the logarithm of the ratio of a term at a given distance \(l\) from the middle. For infinitely large \(n\), this ratio simplifies dramatically to \( e^{-2l^2/n} \), revealing for the first time the underlying continuous distribution. By summing these terms and substituting \( l = s\sqrt{n} \), the paper derives a series that converges to the probability sum. Using this, it calculates specific probabilities: the probability that an event with equal chance will fall within \( \frac{1}{2}\sqrt{n} \) of its expected value is approximately 0.682688; within \( \sqrt{n} \), approximately 0.954288; and within \( \frac{3}{2}\sqrt{n} \), approximately 0.99874. These values represent the first tabulation of what would become the normal probability integral.

Finally, the work generalizes these findings to binomials of the form \((a+b)^n\), where the chances of an event are unequal (ratio \(a:b\)). It demonstrates that the maximum term is approximately \((a+b) / \sqrt{2\pi ab n}\) and that the logarithm of the ratio of any other term to the maximum is \( -\frac{(a+b)^2}{2ab n} l^2 \), thereby extending the approximation to all binomial expansions.

This paper is of unparalleled historical significance. It contains the first derivation and application of the normal distribution curve, the first correct use of an approximation for large factorials (often misattributed as Stirling's formula), and the first practical application of the law of large numbers, providing a precise quantification of probability limits that Bernoulli's theorem could only describe in the abstract. This work marks the birth of modern probability theory and its transition from a combinatorial to a continuous analytical framework.

Abraham de Moivre’s pamphet "Approximatio ad Summam Terminorum Binomii (a + b)ⁿ in Seriem expansi"  was printed on 13 November 1733 for private circulation. A reprint may be found in a paper by R.C. Archibald "A Rare Pamphlet of De Moivre and Some of his Discoveries," Isis, 8 (1926), pp. 671-684.