A233044 Pairs p, q for those partial sums p/q of the series e = sum_{n>=0} 1/n! that are not convergents to e.

1, 1, 5, 2, 65, 24, 163, 60, 1957, 720, 685, 252, 109601, 40320, 98641, 36288, 9864101, 3628800, 13563139, 4989600, 260412269, 95800320, 8463398743, 3113510400, 47395032961, 17435658240, 888656868019, 326918592000

Offset

1,3

Comments

Sondow (2006) conjectured that 2/1 and 8/3 are the only partial sums of the Taylor series for e that are also convergents to the simple continued fraction for e. Sondow and Schalm (2008, 2010) proved partial results toward the conjecture. Berndt, Kim, and Zaharescu (2012) proved it in full.

References

J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part I, in Tapas in Experimental Mathematics, T. Amdeberhan and V. H. Moll, eds., Contemp. Math., vol. 457, American Mathematical Society, Providence, RI, 2008, pp. 273-284.

Links

Table of n, a(n) for n=1..28.

B. Berndt, S. Kim, and A. Zaharescu, Diophantine approximation of the exponential function and Sondow's conjecture, abstract 2012.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly, 113 (2006), 637-641.

J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e?, (and a link to the primes 2, 5, 13, 37, 463), part II, in Gems in Experimental Mathematics, T. Amdeberhan, L. A. Medina, V. H. Moll, eds., Contemp. Math., vol. 517, American Mathematical Society, Providence, RI, 2010, pp. 349-363.

Formula

a(2n-1)/a(2n) = A061354(k)/A061355(k) for some k <> 1 and 3.

a(2n-1)/a(2n) <> A007676(k)/A007677(k) for all k.

Example

1/1, 5/2, 65/24, 163/60, 1957/720, 685/252, 109601/40320, 98641/36288, 9864101/3628800, 13563139/4989600, 260412269/95800320, 8463398743/3113510400, 47395032961/17435658240, 888656868019/326918592000

Crossrefs

Cf. A061354, A061355, A007676, A007677.

Sequence in context: A328555 A208927 A099612 * A142599 A266461 A068566

Adjacent sequences: A233041 A233042 A233043 * A233045 A233046 A233047

Keyword

nonn

Author

Jonathan Sondow, Dec 07 2013

Status

approved