Dances between continuous and discrete: Euler's summation formula

TL;DR

This paper explores Euler's summation formula, its historical development, applications to various mathematical problems like the Basel Problem, and its connection to Bernoulli numbers and factorial approximations.

Contribution

It provides a detailed analysis of Euler's original ideas, applications, and the historical context of his summation formula, including translations and annotations of his work.

Findings

Euler's summation formula estimates sums and diverges almost always.

Application to the Basel Problem and reciprocal squares.

Connection to Bernoulli numbers and Stirling's series.

Abstract

Leonhard Euler likely developed his summation formula in 1732, and soon used it to estimate the sum of the reciprocal squares to 14 digits --- a value mathematicians had been competing to determine since Leibniz's astonishing discovery that the alternating sum of the reciprocal odd numbers is exactly $\pi /4$. This competition came to be known as the Basel Problem, and Euler's approximation probably spurred his spectacular solution in the same year. Subsequently he connected his summation formula to Bernoulli numbers, and applied it to many other topics, masterfully circumventing that it almost always diverges. He applied it to estimate harmonic series partial sums, the gamma constant, and sums of logarithms, thereby calculating large factorials (Stirling's series) with ease. He even commented that his approximation of $ \pi $ was surprisingly accurate for so little work. All this is beautifully presented in mature form in Euler's book Institutiones Calculi Differentialis. I have translated extensive selections for annotated publication as teaching source material in a book Mathematics Masterpieces; Further Chronicles by the Explorers, featuring original sources. I will summarize and illustrate Euler's achievements, including the connection to the search for formulas for sums of numerical powers. I will show in his own words Euler's idea for deriving his summation formula, and how he applied the formula to the sum of reciprocal squares and other situations, e.g., large factorials and binomial coefficients. Finally, I will discuss further mathematical questions, e.g., approximation of factorials, arising from Euler's writings.