Integrals may not help this time...

Can you add up all positive rational numbers less than 1? (Read carefully... I said $rational$ and not $real$ numbers).

#Calculus

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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

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}`	$\boxed{123}$

Comments

Sure they can all be added up. It's an infinite sum. See harmonic sums, like 1/2 + 1/3 + 1/4 + ... etc. Of course, Gaussian rationals can't be included in this sum because "less than 1" has no meaning for complex numbers.

Michael Mendrin - 7 years, 4 months ago

But how?

Maharnab Mitra - 7 years, 3 months ago

I think a more interesting question is, given the set of all positive rationals from 0 to 1, say that the number of the elements in this set is n. Then what is 1/n of the sum of all of them? I don't have an answer to that one yet, I'll think about it when I get the time. I know that (1/n) of the sum 1 + 1/2 + 1/3 + ... + 1/n has the limiting value of 0 as n -> ∞.

Michael Mendrin - 7 years, 4 months ago

Wouldn't this be 1/2 ? We could pair each rational number x (other than 1/2) with (1 - x); then the sum would be lim(n->infinity)[(1/n)(((n-1)/2) + (1/2))] = lim(n->infinity)((1/n)(n/2)) = 1/2.

Brian Charlesworth - 7 years, 4 months ago

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}`	$\boxed{123}$