A374987 Let s(x) be the Maclaurin series for cos(x); then a(n) is the least index k for which all partial sums of cos(2m*Pi) are positive.

6, 14, 24, 32, 40, 48, 58, 66, 74, 82, 92, 100, 108, 116, 126, 134, 142, 150, 160, 168, 176, 184, 194, 202, 210, 218, 228, 236, 244, 254, 262, 270, 278, 288, 296, 304, 312, 322, 330, 338, 346, 356, 364, 372, 382, 390, 398, 406, 416, 424, 432, 440, 450, 458

Offset: 0

Author

Clark Kimberling, Oct 01 2024

Keywords

• nonn

Examples

			For n=1, the partial sums (for k = 0,1,2,3,4,5,6,7) are approximately 1, -18.7, 46.2, -39.2, 20.9, -5.4, 2.4, 0.7; beginning with k=6, the partials sums are all positive, so a(1)=6.
		

Crossrefs

Cf. A376456, A376457, A375057, A375053, A375054.

Programs

Mathematica

z = 800; r = Pi;
f[m_, n_] := f[m, n] = N[Sum[(-1)^k  (2 m  r)^(2 k)/(2 k)!, {k, 0, n}], 10]
g[m_] := Select[Range[z], f[m, #] > 0 && f[m, # + 1] > 0 &, 1]
Flatten[Table[g[m], {m, 1, 80}]]