TITLE: Questions on series
QUESTION [1 upvotes]: We know that if $\sum a_{n}$, $\sum b_{n}$ and $\sum c_{n}$ are convergent series (converging respectively to A,B and C), and if $c_{n}=a_{0}b_{n}+a_{1}b_{n-1}+...+a_{n}b_{0}$, then $C=AB$.
What happens if we do not know if $\sum c_{n}$ converges? I mean, do we have the following theorem also: If $\sum a_{n}$, $\sum b_{n}$  are convergent series (to A and B), and if $c_{n}=a_{0}b_{n}+a_{1}b_{n-1}+...+a_{n}b_{0}$ then $\sum c_{n}$ is a convergent serie and its sum is $AB$. If it is not true in general, then under which condition could it be true?
My second question is: if $\sum_{0}^{n}a_{n}\leq \sum_{0}^{n}b_{n}$, then do we have $A\leq B$ (where $A$ is $\sum a_{n}$ and $B$ is $\sum b_{n}$).
Thank you in advance for your answers

REPLY [1 votes]: For you first question, it is not true in general, take for example $A=B=\sum_n \frac{(-1)^n}{\sqrt{n}}$. The absolute convergence of $A$ or $B$ and the convergence of the other (Mertens theorem) assure that $C$ is convergent.
For your second question, it is effectively true, by just taking limit in both side.
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