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?