Waste less time on Facebook — follow Brilliant.
×

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
1 year, 9 months ago

No vote yet
1 vote

Comments

Sort by:

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 · 1 year, 9 months ago

Log in to reply

@Trung Đặng Đoàn Đức Wow! Awesome!

I also saw this video:

\[ 1+2+3+4+\ldots=\frac{1}{12} \] Adam Phúc Nguyễn · 1 year, 9 months ago

Log in to reply

@Adam Phúc Nguyễn i guess u forgot the negative sign along with 1/12 Kaustubh Miglani · 1 year, 8 months ago

Log in to reply

@Adam Phúc Nguyễn Wow, that's cool :))) Trung Đặng Đoàn Đức · 1 year, 9 months ago

Log in to reply

You forgot \(4^\text{th}\) type of people; they say that this series diverges. Micah Wood · 1 year, 9 months ago

Log in to reply

@Micah Wood Its answer oscillates b/w 0 and 1 Dev Sharma · 1 year, 9 months ago

Log in to reply

@Dev Sharma Yes that's why it diverges. Micah Wood · 1 year, 9 months ago

Log in to reply

Comment deleted Aug 29, 2015

Log in to reply

@Adam Phúc Nguyễn What is that supposed to mean? Micah Wood · 1 year, 9 months ago

Log in to reply

last one is pretty good Kaustubh Miglani · 1 year, 9 months ago

Log in to reply

Gud 1 Naman Kapoor · 1 year, 9 months ago

Log in to reply

Grandi series. Dev Sharma · 1 year, 9 months ago

Log in to reply

haha.. g8 Ankur Bhattacharjee · 1 year, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...