|タイトル||Irrationality and transcendence of alternating series via continued fractions|
|講演者||Jonathan Sondow 氏|
|場所||矢上キャンパス 14棟2階 ディスカッションルーム8|
I give conditions for irrationality of the sum of an alternating series, and introduce a “simple” condition for its transcendence.
The proofs use continued fractions, irrationality measure, and the Thue-Siegel-Roth theorem on rational approximations to algebraic numbers.
The simple continued fractions for an infinite family of naturally-occurring transcendental numbers are given explicitly.
We also prove irrationality and transcendence for families of non-alternating series, using partial sums instead of continued fractions.
Mentioned along the way are Fermat, Fibonacci, and Liouville numbers, π, e, primorials, Sylvester’s sequence, and some conjectures.