OFFSET
1,1
COMMENTS
Let t_1, t_2, ..., t_n be n real numbers in [-1, 1] such that Sum_{i=1..n} (t_i)^k = 0, then lim sup ((Sum_{i=1..n} (t_i)^m)/n) = (x_0)/k, where k > m > 0 are odd numbers and x_0 is the unique positive root of x^(k-m)*(x + k)^m - m^m*(k - m)^(k-m). x_0 is an algebraic integer of degree k - 2. The upper bound (x_0)/k can be approached by x_1 = x_2 = ... = x_(p^k) = -1, x_(p^k+1) = x_(p^k+2) = ... = x_(p^k+q^k) = p/q, where p/q is a rational arbitrarily close to t_0, t_0 is the unique positive root of (k - m)*x^k + k*x^(k - m) - m = 0. Note that (x_0)/k = ((t_0)^m - (t_0)^k)/((t_0)^k + 1).
Here k = 7, m = 1, so the upper bound is (x_0)/7 = 0.57457032651694013806558..., where x_0 = 4.0219922856185809664591... is the unique positive root to x^6*(x + 7) - 6^6 = 0. The upper bound (x_0)/5 can be approached by t_1 = t_2 = ... = t_(p^7) = -1, t_(p^7+1) = t_(p^7+2) = ... = t_(p^7+q^7) = p/q, where p/q is a rational arbitrarily close to t_0, t_0 = 0.67033204760309682774318... is the unique positive root of 6x^7 + 7x^6 - 1 = 0. For example, let p = 67033, q = 100000, t_1 = t_2 = ... = t_(67033^7) = -1, t_(67033^7+1) = t_(67033^7+2) = ... = t_(67033^7+100000^7) = 0.67033, then (Sum_{i=1..67033^7+100000^7} t_i)/(67033^7 + 100000^7) = 0.57457032649925207012850..., very close to (x_0)/7. Note that (x_0)/7 = (t_0 - (t_0)^7)/((t_0)^7 + 1).
LINKS
EXAMPLE
4.0219922856185809664591185627085393007453409109840...
MAPLE
evalf(solve(x^5-5*x^4+24*x^3-108*x^2+432*x-1296=0, x)[1], 120); # Muniru A Asiru, Oct 07 2018
MATHEMATICA
Part[RealDigits[N[Root[x^5 - 5x^4 + 24x^3 - 108x^2 + 432x - 1296, 1], 100]], 1] (* Stefano Spezia, Oct 07 2018 *)
PROG
(PARI) default(realprecision, 100); solve(x=4, 5, x^5 - 5*x^4 + 24*x^3 - 108*x^2 + 432*x - 1296)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Jianing Song, Oct 06 2018
STATUS
approved