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In Equation (0.0.7)-(0.09) of Norbert Wiener's Tauberian Theorems, it is stated that

[...] if $$ \lim_{x\to 1-0}\frac{1}{1-x}\sum_{0}^{\infty}a_nx^n=A, $$ and $$ a_n= o(1/n), $$ then $$ \sum_{0}^{\infty}a_n=A. $$

However, if the last equation holds with $A\not= 0$, then the left hand side term of the first equation is infinite, as far as I can tell. According to Wikipedia, the normalization factor $1/(1-x)$ should be omitted, but I cannot imagine that Wiener put that specific factor there by accident. Is there another way to salvage the equations?

Bananach
  • 7,934
  • Wikipedia's statement should be the correct one as the tauberian theorem is the converse of the abelian theorem "if $\sum_{n=0}^{\infty}a_n=A$ then $\lim_{x\to 1-0}\sum_{n=0}^{\infty}a_nx^n=A$". Another useful case is when $\lim_{x \to 1} (1-x)\sum_{n=0}^{\infty}a_nx^n =A$ and $a_n \ge 0$ then $\lim_{N \to \infty} \frac1N \sum_{n \le N} a_n =A$ – reuns Oct 02 '18 at 19:23

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