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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 \(i\in\mathbb{N}\)?

Thanks in advance for your answers!

Note by Francis Gerard Magtibay
6 months, 1 week ago

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