#### English Title

Introduction to the Analysis of the Infinite, volume 1

#### Enestrom Number

101

#### Fuss Index

87a

#### Original Language

Latin

#### Published as

Quarto book

#### Published Date

1748

#### Written Date

1745

#### Archive Notes

The Euler Archive has a PDF of an 1885 German translation.

#### Content Summary

In the *Introductio in analysin infinitorum* (this volume, together with E102), Euler lays the foundations of modern mathematical analysis. He summarizes his numerous discoveries in infinite series, infinite products, and continued fractions, including the summation of the series 1/1^{k} + 1/2^{k} + 1/3^{k} + ... for all even values of *k* from 2 to 26. Perhaps more importantly, the *Introductio* makes the function the central concept of analysis; Euler introduces the notation *f*(*x*) for a function and uses it for implicit as well as explicit functions, and for both continuous and discontinuous functions. In addition, he calls attention to the central role of *e* and *e*^{x} in analysis. At the end of Chapter 7, he uses infinitesimal analysis to develop the definitions *e ^{x}* = (1 +

*x*/

*i*)

^{i}and ln(1+

*x*) =

*i*((1+

*x*)

^{1/i}– 1), where

*i*represents an infintely small quantity. These formulations put

*e*

^{x}and ln(

*x*) on an equal basis for the first time. Additionally, Euler proves that every rational number can be written as a finite continued fraction and that the continued fraction of an irrational number is infinite. He also shows how infinite series correspond to infinite continued fractions; in particular, Euler derives continued fraction expansions for e and √

*e*.

#### Original Source Citation

Lausanne: Marcum-Michaelem Bousquet, Volume 1, pp. 1-320.

#### Opera Omnia Citation

Series 1, Volume 8, pp.1-392.

#### Record Created

2018-09-25

## Notes

The Euler Archive has a PDF of an 1885 German translation.