A332324 Decimal expansion of the minimum value of the 4th Maclaurin polynomial of e^x.

2, 7, 0, 3, 9, 4, 7, 6, 5, 2, 0, 5, 1, 8, 4, 6, 0, 7, 9, 6, 2, 4, 5, 9, 6, 1, 3, 3, 8, 3, 1, 1, 0, 9, 1, 1, 9, 6, 1, 4, 6, 0, 2, 1, 2, 8, 1, 4, 2, 8, 3, 3, 3, 6, 2, 3, 2, 5, 6, 7, 9, 9, 4, 1, 0, 6, 3, 8, 1, 5, 9, 0, 9, 7, 8, 9, 9, 1, 0, 3, 8, 0, 4, 5, 8, 0

Offset: 0

Author

Clark Kimberling, Feb 11 2020

Keywords

• nonn
• cons
• easy

Comments

Let p(n,x) denote the n-th Maclaurin polynomial of e^x, and let p'(n,x) denote its derivative. Then p'(n+1,x) = p(n,x), so that the real zero of p(n,x), for odd n, is also the value of x that minimizes p(n+1,x). See A117605 for the (negated) real zero p(3,x).

Examples

			Minimum value = 0.2703947652051846079624596133831109119614602128142...
		

Crossrefs

Cf. A117605.

Programs

Mathematica

z = 150; p[n_, x_] := Normal[Series[E^x, {x, 0, n}]];
t = x /. NSolve[p[3, x] == 0, x, z][[1]]
RealDigits[t][[1]]
Plot[Evaluate[p[4, x]], {x, -3, 1}, PlotRange -> {-1, 3}]