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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
2 years, 2 months ago

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Comments

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Top Newest

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 - 2 years, 1 month ago

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Wow! Awesome!

I also saw this video:

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

Adam Phúc Nguyễn - 2 years, 1 month ago

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

Kaustubh Miglani - 2 years, 1 month ago

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Wow, that's cool :)))

Trung Đặng Đoàn Đức - 2 years, 1 month ago

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

Micah Wood - 2 years, 1 month ago

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Its answer oscillates b/w 0 and 1

Dev Sharma - 2 years, 1 month ago

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Yes that's why it diverges.

Micah Wood - 2 years, 1 month ago

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Comment deleted Aug 29, 2015

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What is that supposed to mean?

Micah Wood - 2 years, 1 month ago

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last one is pretty good

Kaustubh Miglani - 2 years, 2 months ago

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Gud 1

Naman Kapoor - 2 years, 1 month ago

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Grandi series.

Dev Sharma - 2 years, 1 month ago

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haha.. g8

Ankur Bhattacharjee - 2 years, 1 month ago

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