I'm currently stuck in Hardy's Book 'Divergent Series' at a very small Proof which i can't seem to grasp.
Here, $(E,q)$ Summation is defined over $\lim_{m \rightarrow \infty} A_m^{(q)}$ or $\sum_{n=0}^\infty a_m^{(q)}$, where
$$ A_m^{(q)} = \sum_{n=0}^m \frac{1}{(q+1)^m} \binom{m}{n} q^{m-n} A_n \\ a_m^{(q)} = \sum_{n=0}^m \frac{1}{(q+1)^m} \binom{m}{n} q^{m-n} a_n $$ where $A_m = \sum_{n=0}^m a_n$. The Operator E is defined over $E a_n = a_{n+1}$.
(8.2.4) is just the definition of $(E,q)$-Summation, while (8.2.5) is $$ (q+1)^{m+1} a_m^{(q)} = (q+E)^m a_0 $$ which follows directly over the Binomial Formula.
Everything is clear to me except the key step from $a_n = (E+q-q)^n a_0$ to $a_n = o\{(q+1)^n + \binom{n}{1} q (q+1)^{n-1} + \cdots + q^n\}$ and am currently feeling very stupid. Any Help appreciated.