# One minus one plus one minus one plus...

There are three types of people in this world:

Evaluate:

$\color{blue}{S}=1-1+1-1+1-1+\ldots$

Type 1

$\color{blue}{S}=(1-1)+(1-1)+(1-1)+\ldots=0+0+0+\ldots=\boxed{0}$

Type 2

$\color{blue}{S}=1-(1-1)-(1-1)-(1-1)-\ldots=1-0-0-0-\ldots=\boxed{1}$

But the $$\displaystyle 3^{rd}$$ type of people did like this:

$1-\color{blue}{S}=1-(1-1+1-1+\ldots)=1-1+1-1+1-1+\ldots = S$

$\Leftrightarrow 1-\color{blue}S=\color{blue}S \Rightarrow 2\color{blue}S=1 \Rightarrow \color{blue}S=\boxed{\frac{1}{2}}$

Note by Adam Phúc Nguyễn
3 years, 4 months 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} $$\boxed{123}$$

Sort by:

You forgot $$4^\text{th}$$ type of people; they say that this series diverges.

- 3 years, 4 months ago

Its answer oscillates b/w 0 and 1

- 3 years, 4 months ago

Yes that's why it diverges.

- 3 years, 4 months ago

Even more here:

Evaluate : $S=1-2+3-4+5-6+ \ldots$

Type 1 : $S=1+(-2+3)+(-4+5)+ \ldots = 1+1+1+1+\ldots=\infty$

Type 2 : $S=(1-2)+(3-4)+(5-6)+ \ldots = -1-1-1-1+\ldots=-\infty$

Type 3 : They go to WolframAlpha, search this:

 1 sum(n from 1 to infty,(-1)^n*n) 

Which shows up that "The ratio test is inconclusive." and "The root test is inconclusive.", from which they implies that the sum is incosistent.

Type 4 : They go to Wikipedia and finds out that the sum is actually equal to $$1/4$$.

- 3 years, 4 months ago

Wow! Awesome!

I also saw this video:

$1+2+3+4+\ldots=\frac{1}{12}$

- 3 years, 4 months ago

Wow, that's cool :)))

- 3 years, 4 months ago

i guess u forgot the negative sign along with 1/12

- 3 years, 4 months ago

last one is pretty good

- 3 years, 4 months ago

haha.. g8

- 3 years, 4 months ago

Grandi series.

- 3 years, 4 months ago

Gud 1

- 3 years, 4 months ago