If $(X, \|\cdot\|)$ is a Banach space then any series that is absolutely convergent must also be convergent. If we take $X$ to be a Frechet space with a countable family of semi-norms $\{P_i\}$. Is it the case that if a sequence $\{x_k\}$ is such that $\sum_k P_i(x_k) < \infty$ for every $i$ then $\sum_k x_k$ must converge to some $x$ in the Frechet topology?
Edit: My original question was written a bit sloppily and was a question about sequences, when I meant it to be a question about series.
In a Banach space $X$, the interesting result is that absolutely convergent series are convergent. That is, if $\sum \|x_n\| < \infty$ then there is $x \in X$ such that $\lim_\limits{n \to \infty} \sum_{k=1}^n x_k = x$. You prove this by checking that if the series is absolutely convergent then its partial sum sequence is Cauchy.
In a Frechet space, the same argument will show that if $\sum_n P_i(x_n) < \infty$ for each $i$ then the partial sum sequence $\big(\sum_{k=1}^n x_k\big)_{n \geq 1}$ is Cauchy for each seminorm $P_i$ and hence is convergent so that again there is $x$ such that $$\lim_\limits{n \to \infty} \sum_{k=1}^n x_k = x.$$
Of course, if you really meant to talk about sequences and not series then if $\sum P_i(x_n) < \infty$ for every $i$ then $x_n \to 0$ for the seminorm $P_i$ for each $i$. In particular, there again exists $x$ such that $x_n \to x$ in $X$. Then a simple argument shows $P_i(x) = 0$ for each $i$ so that $x = 0$. However this is really a less interesting result than the one for series.