Where did the 1 come from?

Consider the following table of numbers:

\[ \large \begin{array} { c | c |c |c |c |c } \, & & & & & \\ \hline & \frac{1}{ 1 \times 2 } & \frac{1}{2 \times 3 } & \frac{1}{3 \times 4} & \frac{1}{4 \times 5} & \ldots \\ \hline & & \frac{1}{2 \times 3 } & \frac{1}{3 \times 4} & \frac{1}{4 \times 5} & \ldots \\ \hline & & & \frac{1}{3 \times 4} & \frac{1}{4 \times 5} & \ldots \\ \hline & & & & \frac{1}{4 \times 5} & \ldots \\ \hline & & & & & \ddots \\ \end{array}\]

If we sum up the first column, we get $\frac{1}{1 \times 2 } = \frac{1}{2}$ .
If we sum up the second column, we get $\frac{2}{2 \times 3 } = \frac{1}{3}$ .
If we sum up the third column, we get $\frac{3}{3 \times 4 } = \frac{1}{4}$ .
This pattern continues, hence the sum of all the terms is

$S = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots$

Now, let's sum up the rows using partial fractions to get a telescoping series.
If we sum up the first row, we get $\displaystyle \sum_{i=1}^\infty \frac{ 1}{ i (i+1) } = 1$ .
If we sum up the second row, we get $\displaystyle \sum_{i=2}^\infty \frac{ 1}{ i (i+1) } = \frac{1}{2}$ .
If we sum up the third row, we get $\displaystyle \sum_{i=3}^\infty \frac{ 1}{ i (i+1) } = \frac{1}{3}$ .
This pattern continues, hence the sum of all the terms is

$S = \color{#D61F06}{1} + \frac{1}{2} + \frac{1}{3} + \ldots$

Where did the $\color{#D61F06} { 1}$ come from?

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}

Easy Math Editor

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

Here's an even simpler example to show why this is flawed: consider the sum $S=1+1+1+\cdots$ What happens when we group it as follows: $S=1+(1+1+1+\cdots)$ That means $S=1+S$ so $0=1$ . something is obviously wrong here.

Daniel Liu - 6 years, 5 months ago

Oh...... Last year there was this note where someone proved that $0 = 1$ .... I think I still get it......

Jeremy Bansil - 6 years, 5 months ago

The trick is that the sum of all the terms $S$ is actually infinite. We can recognize this because we know that the sum of the harmonic series is infinity.

Thus, we are actually saying that $\infty = \infty + 1$ . We cannot conclude that $0 = 1$ , because we do not have additive inverse of infinity, namely that $\infty - \infty \neq 0$ .

Of course, conversely, the above is (can be modified to) a proof by contradiction that the sum of the harmonic series is infinity.

Calvin Lin Staff - 6 years, 5 months ago

@Calvin Lin – Suppose $S=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots$ is finite. Then, by the summations we did above, $S-1 = S$ . Since $S$ is finite, we can subtract $S$ from both sides to get $-1=0$ , contradiction.

Thus, $S$ is infinite.

@Calvin Lin – Yeah, thought so. Nice little note; divergent series are such pesky things.

Jake Lai - 6 years, 5 months ago

Both series diverge so technically they are both positive infinity and are "equal".

Yannick Yao - 6 years, 5 months ago

Actually, you are not exactly dealing with a convergent series.

Agnishom Chattopadhyay - 6 years, 5 months ago

It's because infinity is weird... :)

Bill Huang - 6 years, 5 months ago

WHEN WE SUM UP THE FIRST ROW WE WILL GET 1

Arnab Karmakar - 6 years, 3 months ago

wait S isn't even well defined

Bill Huang - 6 years, 2 months ago

I am fairy certain the top row does not approach 1... More like it approaches 0.75, The second row does not approach 1/2, more like it approaches 0.2 (both of these are quick evaluations and would require more time to evaluate) but I think the math holds up, just not the assumptions.

Rick Thomas - 4 years, 8 months ago

Step 1: Assume: 1/2+1/3+1/4+... converges: Then: [insert original post here] Which is a contradiction. Thus, 1/2+1/3+... diverges

Terence Coelho - 4 years, 8 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}`	$\boxed{123}$