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:

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

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

## Comments

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TopNewestEven 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: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 · 1 year, 6 months agoLog in to reply

I also saw this video:

\[ 1+2+3+4+\ldots=\frac{1}{12} \] – Adam Phúc Nguyễn · 1 year, 6 months ago

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– Kaustubh Miglani · 1 year, 6 months ago

i guess u forgot the negative sign along with 1/12Log in to reply

– Trung Đặng Đoàn Đức · 1 year, 6 months ago

Wow, that's cool :)))Log in to reply

You forgot \(4^\text{th}\)

typeof people; they say that this series diverges. – Micah Wood · 1 year, 7 months agoLog in to reply

– Dev Sharma · 1 year, 6 months ago

Its answer oscillates b/w 0 and 1Log in to reply

– Micah Wood · 1 year, 6 months ago

Yes that's why it diverges.Log in to reply

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– Micah Wood · 1 year, 6 months ago

What is that supposed to mean?Log in to reply

last one is pretty good – Kaustubh Miglani · 1 year, 7 months ago

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Gud 1 – Naman Kapoor · 1 year, 7 months ago

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Grandi series. – Dev Sharma · 1 year, 7 months ago

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haha.. g8 – Ankur Bhattacharjee · 1 year, 7 months ago

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