### Abstract

We begin by examining a hitherto unexamined partial manuscript by Ramanujan on the diophantine approximation of ^{e2/a} published with his lost notebook. This diophantine approximation is then used to study the problem of how often the partial Taylor series sums of e coincide with the convergents of the (simple) continued fraction of e. We then develop a p-adic analysis of the denominators of the convergents of e and prove a conjecture of J. Sondow that there are only two instances when the convergents of the continued fraction of e coalesce with partial sums of e. We conclude with open questions about the zeros of certain p-adic functions naturally occurring in our proofs.

Original language | English (US) |
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Pages (from-to) | 1298-1331 |

Number of pages | 34 |

Journal | Advances in Mathematics |

Volume | 248 |

DOIs | |

State | Published - Nov 25 2013 |

### Keywords

- Approximation by partial sums of power series
- Continued fractions
- Diophantine approximation
- Exponential function
- P-Adic analysis
- Primary
- Secondary
- Sondow's Conjecture
- Supercongruences

### ASJC Scopus subject areas

- Mathematics(all)