First note that you cannot mean that all subsequences converge to the same limit, yet the series does not converge, since the entire series is a subseries of itself.
It isn't clear what you want in your sequence, but here is an example of a series having a countable collection of subseries $A_1,A_2,...$, with no two terms equal in the entire series, but for which each subseries $A_k$ converges to $0$.
First we need the notion of a squarefree number, which may be defined as any number whose prime factorization has each prime to the power 1. For example since $15=3 \cdot 5$, $15$ is squarefree, while since $12=2\cdot 2 \cdot 3$ has the repeated prime $2$, $12$ is not squarefree.
Now define the sequence $x_n$ to be $n$ if $n$ is not squarefree, and $1/n$ if $n$ is squarefree. Further for each $k \ge 1$ define $A_k$ to be the set of squarefree numbers having exactly $k$ prime factors. For example $A_1$ is the sequence of primes, $A_2$ is the sequence of numbers (put into increasing order) of the form $pq$ where $p,q$ are distinct primes, and so on.
Then each subsequence $A_k$ converges to $0$, and there are countably many such subsequences $A_k$. The series itself does not converge, since it has the subsequence $2^2,3^2,4^2,...$ which diverges to $+\infty.$