Are there more convergent or divergent series? Has the question itself any meaning? (I couldn't phrase that in a rigorous way)

 Are there more convergent or divergent series? Has the question itself any meaning? (I couldn't phrase that in a rigorous way)👇🏾👇🏾👇🏾

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The question of whether there are more convergent or divergent series touches upon concepts from both real analysis and set theory, particularly the theory of infinite sets and cardinality. Let's break it down into a more rigorous analysis.

Convergent vs. Divergent Series

A series $\sum {𝑎}_{𝑛}$ is called convergent if the sequence of its partial sums $({𝑆}_{𝑛})$, where ${𝑆}_{𝑛}={\sum }_{𝑘=1}^{𝑛}{𝑎}_{𝑘}$, converges to a finite limit $𝐿$. If the sequence of partial sums does not converge, the series is called divergent.

Cardinality of Sets

To compare the number of convergent and divergent series, we consider the set of all possible series, which involves considering all possible sequences $({𝑎}_{𝑛})$.

1. Set of All Sequences: The set of all sequences of real numbers is uncountable. This can be shown by a diagonal argument similar to Cantor's argument for the uncountability of the real numbers.

2. Set of Convergent Series: The set of convergent series is also uncountable. For example, consider the geometric series with ${𝑎}_{𝑛}={𝑟}^{𝑛}$ for $𝑟$ such that $\mathrm{\mid }𝑟\mathrm{\mid }<1$. Each value of $𝑟$ within this interval gives a unique convergent series, and there are uncountably many such values of $𝑟$.

3. Set of Divergent Series: Since the set of all series is uncountable and the set of convergent series is a subset of this, the set of divergent series is also uncountable. Furthermore, the divergent series form a larger "type" of uncountable set in a certain sense because the conditions for divergence are less restrictive than for convergence.

Measure Theory Perspective

From a measure theory perspective, we can consider the "size" of sets in terms of measure. Within the space of all possible sequences, the set of convergent series is a "small" set in the sense that it is a meager set (a countable union of nowhere dense sets). In contrast, the set of divergent series is the complement of this and is "large" in the sense that it is comeager (contains a dense open set).

Conclusion

In summary, while both the sets of convergent and divergent series are uncountable, the set of divergent series is, in some sense, "larger." This is because:

• The conditions for divergence are less restrictive, making the divergent series more prevalent within the space of all possible series.
• From a measure theory perspective, the set of divergent series is comeager, indicating it is the "larger" set in a topological sense.

Thus, while both sets are uncountable, there are "more" divergent series in the sense that they form a larger part of the space of all possible series.