Find the sum of the series 1,-1,1,-1,1,-1,...

[vikasj007]

sum of series?

find the sum of the series

1,-1,1,-1,1,-1,...


does it equal 1 or 0?

 

[Curious3141]

As an infinite series, it is not convergent since the "final" term is non-zero. So the sum is not defined.

As a finite series, the first term and the number of terms are defined, so there is no ambiguity.

 

[RandallB]

Would it be fair to say that the AVERAGE of the sum of the series - as the sum of the series is being totaled would alternate between Zero and a number trending towards Zero.

And the final average value would trend together to a value when multiplied by of the infinite number of units in the series would give Zero ??

 

[Gokul43201]

No, it would not be.

The sum just does not exist if the series doesn't converge. So it is meaningless to talk of the "average of the sum" (and I'm not even sure what you mean by that...what is the average of a number ?)

 

[RandallB]

A number by it self cannot have an average.

The Average of a sum, is that sum divided by the number of units or elements that were used to come to that sum.

SUMv (for n = even # series) = Zero
e.g. SUMv (n=10) = 0 AVG SUMv (when n even) = 0 i.e.. 0/10 = 0
When n is even AVG SUMv = 0

SUMv (for n= odd #) = 1
e.g.. n=11 SUMv (n=11) = 1 AVG SUMv (n=11) = 1/11
When n is odd AVG SUMv converges on ZERO as n goes to infinite

Therefore AVG SUMv also converges on Zero.
For the vikasj007 series; AKA v

Limited to answering the “brain teaser” question of:

“ does it equal 1 or 0? “

I’ll go with:

...a) 0

...b) yes
or
...c) Question is non sequitur

 

[Gokul43201]

A number by it self cannot have an average.

The Average of a sum, is that sum divided by the number of units or elements that were used to come to that sum.

If you mean the average value of the terms in the sequence, yes, you are right. The average goes to 0 as the number of terms tends to infinity.

But this doesn't say anything about the sum itelf. The only thing that can be said about the sum is that it is 1 if the number of terms is odd, 0 if the number of terms is even, and not defined if the number of terms is infinite.

 

[RandallB]

it is 1 if the number of terms is odd, 0 if the number of terms is even, and not defined if the number of terms is infinite.

So you want to go with "Question is non sequitur" ?

Personally I kind of like "Yes" as the best answer.

 

[Gokul43201]

I like "no" better...but maybe I'm just a meanie.

 

[RandallB]

does it equal 1 or 0?

OK vikasj - It's your teaser - did you have a conclusion in mind?

 

[vikasj007]

OK vikasj - It's your teaser - did you have a conclusion in mind?

well not really, actually it was a question that came up in the newspaper(i think so), and they had invited replies for it. though some people had replied that a conclusive solution to this is not possible, and some even suggested their theories to come to an answer, but a desicive answer was not given so i decided to put this up to see if i can get some answers.

and i must say you did try. THANKS A LOT!

 

[Gokul43201]

and i must say you did try. THANKS A LOT!

We did better. We gave you the decisive answer. In post #2 and the second half of post #7, you will find the solution to the teaser.

There's absolutely no ambiguity about it.