E684 -- De radicibus aequationis infinitae 0 = 1 - (xx)/(n(n+1)) + (x^4)/(n(n+1)(n+2)(n+3))

E684 -- De radicibus aequationis infinitae \(0 = 1 - \frac{xx}{n(n+1)} + \frac{x^4}{n(n+1)(n+2)(n+3)} - \frac{x^6}{n\cdot\cdots\cdot (n+5)} + \text{etc.}\)

(On the roots of the infinite equation \(0 = 1 - \frac{xx}{n(n+1)} + \frac{x^4}{n(n+1)(n+2)(n+3)} - \frac{x^6}{n\cdot\cdots\cdot (n+5)} + \text{etc.}\))

Summary:

The formula has different roots depending on the parameter n. Sometimes, it behaves like cos(x), others like sin(x)/x, etc.

Publication:

Originally published in Nova Acta Academiae Scientarum Imperialis Petropolitinae 9, 1795, pp. 19-40
Opera Omnia: Series 1, Volume 16, pp. 241 - 265

Documents Available:

Original publication: E684

Return to the Euler Archive