×

# 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
1 year ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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} $$\boxed{123}$$