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# Question about the Comparison Test

I've been curious about this for a while now, but I'm not really good at topics related to infinite series, so I'll post the question here.

The Comparison Test is a useful tool for determining the convergence or divergence of infinite series. It can be stated as follows:

Suppose we have two infinite series $${a_n}$$ and $${b_n}$$ such that $$a_i<b_i$$ for all $$i\in\mathbb{N}$$.

If $${a_n}$$ is divergent, so is $${b_n}$$.

If $${b_n}$$ is convergent, so is $${a_n}$$.

Question: Is there an infinite series $${c_n}$$ that acts as a "boundary" between convergent and divergent series?

In math-speak, does there exist an infinite series $${c_n}$$ such that, for all infinite series $${a_n}$$ and $${b_n}$$:

$${a_n}$$ is convergent iff $${a_i\leq c_i}$$ for all $$i\in\mathbb{N}$$, and

$${b_n}$$ is divergent iff $${b_i\geq c_i}$$ for all $$i\in\mathbb{N}$$?

Note by Francis Gerard Magtibay
6 months, 1 week ago