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Calculus Level 2

What is the value of $1-1+1-1+1-1+ \cdots?$

The series does not converge 1 0

\dfrac{1}{2}

2 solutions

Margaret Zheng
Jan 23, 2016

As this summation grows, its value forms another series: 1,0,1,0,1,0,1... This series does not have any limit, thus the summation does not converge.

Your solution say first option is the answer.

If we do like 1-(1-1)-(1-1)... We may see 1 as answer .

if we do like (1-1)+(1-1)... we may say zero as answer.

if we do something like

S=1-1+1-1.....

S=1-(1-1+1-1...)=1-S

That is S=1/2.

but of course you are right.

Aakash Khandelwal - 5 years, 4 months ago

Isn't that amazing??

Aakash Khandelwal - 5 years, 4 months ago

Could somebody please help me rewrite this solution? I understand the concept but could not explain it well :(

Margaret Zheng - 5 years, 4 months ago

You may refer to the wiki I wrote: LINK

Nihar Mahajan - 5 years, 4 months ago

Thank you VERY much and now I feel very clear about it!

Margaret Zheng - 5 years, 4 months ago

@Margaret Zheng – No worries , it was my pleasure :)

Nihar Mahajan - 5 years, 4 months ago

Did u c the video that numberphile posted on their channel on YouTube..https://youtu.be/PCu_BNNI5x4...above is the link..please explain this 2.. @Calvin Lin

Righved K - 5 years, 4 months ago

I am not getting the option of writing the answer on brilliant.org, hence I am writing in the comment reply.

I would like to write a Note on this question and answrs,

The series does not converse and accumulate at either 1 or 0 that is true but it is not limited to that, even you can rearrange in such a way that it can accumulate at 5 and 3 also as shown below

1+1+1+1+1-1-1+1+1-1-1+1+1-1-1+1+1.......

and same way you can set that series in such a way that it can accumulate to other digits also.

Grandi Series: Surprisingly nobody has mentioned Grandi series, this question is a Grandi series, probably @Righved K has mentioned about the Youtube channel Numberphile video hence that video has given the details of the Grandi series. In year 1703 Italian mathematician Grandi has proved that 1-1+1-1+1....... = 1/2 and it has been accepted by the mathematicians and have been used to prove other theories and other proofs hence this answer can not be denied.
Current Question & its options: Now if I have above both facts I can not deny the option 1/2, which is given in the option of the question, hence person who choose that -The series does not converge and -1/2 are both true, because both are proven things.

Reference for all above facts

Link of Numberphile video

Hence now question itself creates confusion, All given answers are correct in their term.

Option - 1 : It accumulates at 1-That is true, if you rearrange in one way.

Option - 0 : It accumulate at 0-That is also true, if we arrange the numbers in other way.

Option - The series does not converge : That is also true, No doubt

Option - 1/2 : That is also true, If you go by Grandi's series.

So, I am not understanding why this question and its option are given in such a way. Moreover this question with this options are not a candidate of MCQ questions. It is a Subjective question, more discussion type.

Imran Ansari - 5 years, 3 months ago

Imran is correct - this is Grandi's series. It is, counterintuitively, both a divergent series and one that sums to 1/2. Both answers are correct.

Christopher Williams - 5 years, 3 months ago

Congrats Christopher, you have understanding of this concept!!!!

Imran Ansari - 5 years, 3 months ago

S=1-(1-1+1-1...)=1-S is S=1-0=1-S. S=1=1-S If S is one half, 1/2=1=1/2 Not same

. . - 10 months, 4 weeks ago

Sudhanshu Mishra
Jan 31, 2016

There is not given that the series is infinit or any more.. thats why u can't say s=1-s

Let A=1-1+1-1+1-1+....... Add -1 both side A-1= -1+(1-1+1-1+1-1+....) A-1=0-1+1-1+1-1+..... A-1= -(1-1+1-1+1-1+....) A-1= -A 2A=1 A=1/2 1-1+1-1+......=1/2

kumar k - 2 years, 9 months ago