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 - (xx)/(n(n+1)) + (x⁴)/(n(n+1)(n+2)(n+3)) - (x⁶)/(n.....(n+5)) + etc.

(On the roots of the infinite equation 0 = 1 - (xx)/(n(n+1)) + (x⁴)/(n(n+1)(n+2)(n+3)) - (x⁶)/(n.....(n+5)) + 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

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E684 -- De radicibus aequationis infinitae 0 = 1 - (xx)/(n(n+1)) + (x4)/(n(n+1)(n+2)(n+3)) - (x6)/(n.....(n+5)) + etc.

E684 -- De radicibus aequationis infinitae 0 = 1 - (xx)/(n(n+1)) + (x⁴)/(n(n+1)(n+2)(n+3)) - (x⁶)/(n.....(n+5)) + etc.