A one-sided Tauberian theorem for the Borel summability method

We establish a quantitative version of Vijayaraghavan’s classical result and use it to give a short proof of the known theorem that a real sequence (sn) which is summable by the Borel method, and which satisfies the one-sided Tauberian condition that √n(sn − sn−1) is bounded below must be convergent.  2004 Elsevier Inc. All rights reserved