## E616 -- De transformatione seriei divergentis $$1 - mx + m(m+n)x^2 - m(m+n)(m+2n)x^3 + m(m+n)(m+2n)(m+3n)x^4 \text{ etc.}$$ in fractionem continuam

(On the transformation of the divergent series $$1 - mx + m(m+n)x^2 - m(m+n)(m+2n)x^3 + m(m+n)(m+2n)(m+3n)x^4 \text{ etc.}$$ into a continued fraction)

Summary:

In this paper, Euler transforms the divergent series in the title, and thereby dervies a continued fraction expansion for $$\frac{\pi}{4}$$ as

$$\large \displaystyle \frac{\pi}{4} \;=\; \frac{1}{1+\frac{1}{2+\frac{9}{2+\frac{25}{2+\frac{49}{2+\frac{81}{2+\cdots}}}}}}$$

Publication:
• Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 2, 1788, pp. 36-45
• Opera Omnia: Series 1, Volume 16, pp. 34 - 46
Documents Available:
• Original Publication: E616
• Ed Sandifer has made a translation available.
• German translation (Artur Diener and Alexander Aycock): E616

• The Euler Archive attempts to monitor current scholarship for articles and books that may be of interest to Euler Scholars. Selected references we have found that discuss or cite E616 include:
• Dudley RM., “Some inequalities for continued fractions.” Mathematics of Computation, 49 (180), pp. 585-593 (Oct 1987).