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A154310
Decimal expansion of convergent sum of weighted self-defining reciprocals.
0
1, 5, 1, 8, 7, 3, 7, 2, 4, 7, 4, 7, 7, 9, 0, 3, 9, 1, 4, 7, 4, 4, 2, 9, 8, 7, 5, 0, 1, 7, 6, 8, 0, 5, 1, 3, 4, 3, 9, 6, 5, 2, 3, 3, 5, 3, 3, 9, 0, 3, 3, 4, 6, 9, 5, 9, 9, 3, 9, 9, 9, 9, 1, 0, 0, 4, 5, 7, 3, 1, 7, 8, 9, 9, 6, 9, 6, 4, 8, 3, 4, 4, 4, 4
OFFSET
1,2
COMMENTS
Let x(1)=1, let x(2) be x>0 satisfying 1/x(1) + 1/x = x,..., for n>=1, let x(n+1) be x>0 satisfying SUM(1/x(k): k=1,2,...,n) + 1/x = x.
Then SUM(1/x(k): k=1,2,...) diverges, but if the k-th term is replaced by w(k)/x(k) where w(k)=2^(1-k), the resulting sum converges to S=1.518737... .
x(1)=1, x(2)=1.618... (the golden ratio), x(3)=2.095293... .
LINKS
Clark Kimberling, Polynomials associated with reciprocation, Journal of Integer Sequences 12 (2009, Article 09.3.4) 1-11.
EXAMPLE
1.51873724747790391474429875017680513439652335339033...
CROSSREFS
Sequence in context: A201525 A269229 A193089 * A193856 A255294 A115521
KEYWORD
nonn,cons
AUTHOR
STATUS
approved