A095822 Numerators of certain upper bounds for Euler's number e.

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

Links

Table of n, a(n) for n=1..19.

W. Lang, r(n) numbers and comments.

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.

Cf. A001339, A001563.

Sequence in context: A187249 A105151 A111680 * A025539 A172440 A254536

Adjacent sequences: A095819 A095820 A095821 * A095823 A095824 A095825

Keyword

nonn,easy,frac

Author

Wolfdieter Lang, Jun 11 2004

Status

approved