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Suppose one has an infinite series of positive reals. And suppose that there are an infinite number of sets of terms $S_n$ which take the same value. That is:

$S_1 = \{ a_{1,1}, a_{1,2},...,a_{1,m}\}$, $a_{1,m} = c_1$

$S_2 = \{ a_{2,1}, a_{2,2},...,a_{2,m}\}$, $a_{2,m} = c_2$

...

$S_n = \{ a_{n,1}, a_{n,2},...,a_{n,m}\}$, a_{n,n} = c_m$

And the cardinality of each $S_n$ is finite, but increases exponentially. Will a series with these properties always converge? Always diverge?

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    $\begingroup$ Sometimes converge, sometimes not. $\endgroup$ – André Nicolas Feb 20 '16 at 23:18
  • $\begingroup$ Do you mean series or sequence? The sequence might converge if, say, the $c_i$ converge. $\endgroup$ – lulu Feb 20 '16 at 23:18
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Of course the answer is, it depends. Roughly speaking, it depends on the comparison between the rates of growth of $\vert S_i\vert$ and the rate of shrinkage of $c_m$. If the $c_m$s shrink "much faster" than the sizes of the $S_i$s grow, then the corresponding series will converge; otherwise, it will diverge. The rate of growth of $\vert S_i\vert$ on its own tells you absolutely nothing.

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