OFFSET
1,1
COMMENTS
Conjecture: a(n) is finite for every positive integer n, and a(n) approaches infinity with n.
EXAMPLE
Let H(k) = Sum_{j=1..k} 1/j, i.e., the k-th harmonic number.
a(1)=2: Newton's method applied to f(x)=exp(-1-x) starting at x(1)=1 gives x(1)=1, x(2)=2, whereas H(1)=1 and H(2)=1.5.
a(3)=11: Newton's method applied to f(x) = exp(-1 - x - x^2/2 - x^3/6) starting at x(1)=1 gives x(10)=2.91631 and x(11)=3.03873, whereas H(10)=2.92897 and H(11)=3.01988.
PROG
(PARI)
a(n)=
{default(seriesprecision, n);
default(realprecision, max(38, 2*#digits(n!, 10)));
Tp = truncate(taylor(exp(x), x, n));
t=1.0;
hsum=1.0;
j=1;
while(t<=hsum, j=j+1; t=t+1.0/subst(Tp, x, t); hsum = hsum + 1.0/j);
j; }
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jennifer Sirkin, Oct 01 2021
EXTENSIONS
a(15) inserted by Jinyuan Wang, Dec 10 2021
STATUS
approved