One minus one plus one minus one plus...

There are three types of people in this world:

Evaluate:

$\color{#3D99F6}{S}=1-1+1-1+1-1+\ldots$

Type 1

$\color{#3D99F6}{S}=(1-1)+(1-1)+(1-1)+\ldots=0+0+0+\ldots=\boxed{0}$

Type 2

$\color{#3D99F6}{S}=1-(1-1)-(1-1)-(1-1)-\ldots=1-0-0-0-\ldots=\boxed{1}$

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

$1-\color{#3D99F6}{S}=1-(1-1+1-1+\ldots)=1-1+1-1+1-1+\ldots = S$

$\Leftrightarrow 1-\color{#3D99F6}S=\color{#3D99F6}S \Rightarrow 2\color{#3D99F6}S=1 \Rightarrow \color{#3D99F6}S=\boxed{\frac{1}{2}}$

#Algebra

Easy Math Editor

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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

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

Micah Wood - 5 years, 9 months ago

Its answer oscillates b/w 0 and 1

Dev Sharma - 5 years, 9 months ago

Yes that's why it diverges.

Micah Wood - 5 years, 9 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 $1/4$ .

Trung Đặng Đoàn Đức - 5 years, 9 months ago

Wow! Awesome!

I also saw this video:

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

Adam Phúc Nguyễn - 5 years, 9 months ago

Wow, that's cool :)))

Trung Đặng Đoàn Đức - 5 years, 9 months ago

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

Kaustubh Miglani - 5 years, 8 months ago

last one is pretty good

Kaustubh Miglani - 5 years, 9 months ago

haha.. g8

Ankur Bhattacharjee - 5 years, 9 months ago

Grandi series.

Dev Sharma - 5 years, 9 months ago

Gud 1

Naman Kapoor - 5 years, 9 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}`	$\boxed{123}$