A095822 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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.

Wolfdieter 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