One minus one plus one minus one plus...

There are three types of people in this world:

Evaluate:

S=11+11+11+ \color{#3D99F6}{S}=1-1+1-1+1-1+\ldots

Type 1

S=(11)+(11)+(11)+=0+0+0+=0 \color{#3D99F6}{S}=(1-1)+(1-1)+(1-1)+\ldots=0+0+0+\ldots=\boxed{0}

Type 2

S=1(11)(11)(11)=1000=1 \color{#3D99F6}{S}=1-(1-1)-(1-1)-(1-1)-\ldots=1-0-0-0-\ldots=\boxed{1}

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

1S=1(11+11+)=11+11+11+=S 1-\color{#3D99F6}{S}=1-(1-1+1-1+\ldots)=1-1+1-1+1-1+\ldots = S

1S=S2S=1S=12 \Leftrightarrow 1-\color{#3D99F6}S=\color{#3D99F6}S \Rightarrow 2\color{#3D99F6}S=1 \Rightarrow \color{#3D99F6}S=\boxed{\frac{1}{2}}

Note by Adam Phúc Nguyễn
3 years, 11 months ago

No vote yet
1 vote

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Comments

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Top Newest

You forgot 4th4^\text{th} type of people; they say that this series diverges.

Micah Wood - 3 years, 11 months ago

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Its answer oscillates b/w 0 and 1

Dev Sharma - 3 years, 11 months ago

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Yes that's why it diverges.

Micah Wood - 3 years, 11 months ago

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Even more here:

Evaluate : S=12+34+56+S=1-2+3-4+5-6+ \ldots

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

Type 2 : S=(12)+(34)+(56)+=1111+=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 1/41/4.

Trung Đặng Đoàn Đức - 3 years, 11 months ago

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Wow! Awesome!

I also saw this video:

1+2+3+4+=112 1+2+3+4+\ldots=\frac{1}{12}

Adam Phúc Nguyễn - 3 years, 11 months ago

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Wow, that's cool :)))

Trung Đặng Đoàn Đức - 3 years, 11 months ago

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

Kaustubh Miglani - 3 years, 11 months ago

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last one is pretty good

Kaustubh Miglani - 3 years, 11 months ago

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haha.. g8

Ankur Bhattacharjee - 3 years, 11 months ago

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Grandi series.

Dev Sharma - 3 years, 11 months ago

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Gud 1

Naman Kapoor - 3 years, 11 months ago

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