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 $i\in\mathbb{N}$ ?

Thanks in advance for your answers!

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$