I wonder what differences are between the folowwing two versions of Abel's criteria for uniform convergence:
From Elementary classical analysis by Marsden and Hoffman:
Abel's Test. Let $A \subset R^n$ and $\phi_n: A \rightarrow R$ be a sequence of functions which are decreasing; that is, $\phi_{n+1}(x) \leq \phi_n(x)$ for each $x \in A$. Suppose there is a constant $M$ such that$|\phi_n(x)| \leq M$ for all $x \in A$ and all $n$. If $\displaystyle\sum_{n=1}^\infty f_n(x)$ converges uniformly on $A$, then so does $\displaystyle \sum_{n=1}^\infty \phi_n(x)f_n(x)$.
From Wikipedia:
Abel's uniform convergence test. Let $\{g_n\}$ be a uniformly bounded sequence of real-valued continuous functions on a set $E$ such that $g_{n+1}(x) \leq g_n(x)$ for all $x ∈ E$ and positive integers $n$, and let $\{f_n\}$ be a sequence of real-valued functions such that the series $\displaystyle\sum f_n(x)$ converges uniformly on $E$. Then $\displaystyle\sum f_n(x)g_n(x) $converges uniformly on $E$.
- Is the additional requirement of continuity for a sequence of functions in Wikipedia the only difference? If not, what else?
- Is this continuity unnecessary and can be ignored as in Marsden's? If yes, is Marsden's a more general version? Or do you have a different one?
Thanks and regards!