2

Does anyone know of a simple proof that $\sum_{n=0}^{\infty}a_{n}$ converges whenever the real sequence $\{ a_{n} \}_{n=0}^{\infty}$ satisfies these two conditions?

  1. $a_{n}=O(1/n)$;
  2. $\lim_{x\uparrow 1}\sum_{n=0}^{\infty}a_{n}x^{n}$ exists.

I believe this is known as the Hardy Tauberian Theorem.

Disintegrating By Parts
  • 87,459
  • 5
  • 65
  • 149
  • 1
    You can find proofs to various Tauberian theorems (the theorem you are after is called Littlewood’s Theorem in the note) in this note. – Winther Jan 28 '15 at 19:15
  • @Winther : Thank you for that PDF. The proof there is 6+ pages long and uses Weierstrass approximation. That's not so simple, but the fact that your PDF is not ancient, and is full of cleverness makes me think that may be the state-of-art proof. – Disintegrating By Parts Jan 28 '15 at 19:56
  • 1
    You can also check out this MSE question which has a good answer with some references (the PDF I linked to turns out to be from the book mentioned in that answer). – Winther Jan 28 '15 at 20:31
  • @Winther : That's a great discussion. I upped your answer and took note of the full book. Interesting. Based on that, I'm betting there is no simpler proof known at this point than Karamata's. Some things just can't be simple. – Disintegrating By Parts Jan 28 '15 at 20:46

0 Answers0