I am studying some material on real analysis. The word "summable" troubled me. I searched online. It looks like summable means having a Lebesgue integral. But Lebesgue integral is not a familiar thing, either. I wonder is there a definition or interpretation of summable that could avoid Lebesgue integral. Thank you so much for your help! Really appreciate it!
$\endgroup$1 Answer
$\begingroup$Let ${a_n}$ be a sequence, and let $s_k=a_1+a_2+...+a_k$
be the $k^{th}$ partial sum of the series $\sum_{n=1}^\infty a_n.$
The series $\sum_{n=1}^\infty a_n$ is called summable, with sum $A \in \mathbb R$, if the average value of its partial sums $s_k$ tends to $A$ i.e. $lim_{n\rightarrow \infty}1/n \sum_{k=1}^n s_k=A$
$\endgroup$ 1
