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A095822
Numerators of certain upper bounds for Euler's number e.
2
3, 11, 49, 87, 1631, 11743, 31967, 876809, 8877691, 4697191, 1193556233, 2232105163, 2222710781, 3317652307271, 53319412081141, 303328210950491, 2348085347268533, 313262209859119579, 42739099682215483
OFFSET
1,1
COMMENTS
e = Sum_{k>=0} 1/k! has upper bound r(n) = a(n)/A095823(n). See the W. Lang link.
REFERENCES
M. Barner and F. Flohr, Analysis I, de Gruyter, 5te Auflage, 2000; pp. 117/8.
E. Kuz'min and A. I. Shirshov: On the number e, pp. 111-119, eq.(6), in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am.Math.Soc., 1999
FORMULA
a(n) = numerator(r(n)), with rational r(n) = Sum_{k=0..n} 1/k! + 1/(n*n!), n>=1, written in lowest terms. For n*n! see A001563(n).
From Peter Bala, Oct 08 2019: (Start)
r(n) = 3 - 1/(4 - 2/(5 - 3/(6 - ... - (n-1)/(n+2)))).
r(n) = 3 - Sum_{k = 2..n} 1/(k!*k*(k - 1)).
r(n) = (1/(n*n!))*Sum_{k = 0..n} (k+1)!*binomial(n,k) = A001339(n)/A001563(n).
r(n) = r(n-1) - 1/(n!*n*(n-1)) for n >= 2. (End)
r(n) = ((n+1)/n)*hypergeom([-n], [-n-1], 1). - Peter Luschny, Oct 09 2019
EXAMPLE
The positive rationals r(n), n>=1: 3/1, 11/4, 49/18, 87/32, 1631/600, 11743/4320, 31967/11760, ...
MATHEMATICA
r[n_] := ((n + 1)/n) HypergeometricPFQ[{-n}, {-n - 1}, 1];
Table[Numerator[r[n]], {n, 1, 19}] (* Peter Luschny, Oct 09 2019 *)
CROSSREFS
The denominators are in A095823.
Sequence in context: A187249 A105151 A111680 * A025539 A172440 A254536
KEYWORD
nonn,easy,frac
AUTHOR
Wolfdieter Lang, Jun 11 2004
STATUS
approved