To read Method One click here.

What is \(1 - 2 + 3 - 4 + 5 - ...\) equal?

Let's call the sum \(s\).

\(2s = (1 - 2 + 3 - 4 + ...) + (1 - 2 + 3 - 4 + ...)\)

\(2s = 1 + (- 2 + 3 - 4 + 5 - ...) + 1 - 2 + (3 - 4 + 5 - 6 + ...)\)

\(2s = 0 + (- 2 + 3) + (3 - 4) + (- 4 + 5) +...\)

\(2s = 1 - 1 + 1 - 1 + 1 ...\)

\(2s = 1 + (- 1 + 1) + (- 1 + 1) + ...\)

\(2s = 1 + 0 + 0 + 0 + ...\)

\(s = \frac {1}{2}\)

But, from my previous method, the answer was \(\frac {1}{4}\)!

How is this possible?

## Comments

Sort by:

TopNewestNote that \(1-1+1-1+\cdots=\dfrac{1}{2}\), via another famous summation. Therefore \(2s=\dfrac{1}{2}\), and \(s=\dfrac{1}{4}\) as desired. \(\Box\) – Daniel Liu · 3 years, 4 months ago

Log in to reply

This is because of Rienmann series thereom. This sequence is conditionally convergent so rienmann series theorem states that it could be rearranged in order to get any number. http://en.wikipedia.org/wiki/Riemann

seriestheorem – Ashar Tafhim · 3 years, 6 months agoLog in to reply

There are two possibilities:1)the one shown in the note.2)the,series of( 1s )(-1s) ending with a -1 then the answer would be 0. – Adarsh Kumar · 3 years, 2 months ago

Log in to reply

Again, using the same method in your previous link that I mentioned, the partial sums diverge, so the sum also diverges. The partial sums for the first 2k terms is -1, for the first 2k+1 terms is (k+1)/2. Clearly the partial sums diverge. – Yong See Foo · 3 years, 6 months ago

Log in to reply

– Sharky Kesa · 2 years, 2 months ago

Hey Yong See, you remember me from camp?Log in to reply

What you have done here is mathematically incorrect. You can't pair up terms like that. As a matter of fact on the line \(2s=1-1+1-1+1+...+(-1)^n\) is a geometric series with \(r=-1\) and the sum diverges, i.e. it does not sum up to a specific value (A geometric series only converges when \(|r|<1\)). The correct method to solve this problem is to pair up the terms as:

\(s = 1-2+3-4+5...=\lim_{n \rightarrow \infty}(1+3+5+...+(2n+1))-(2+4+6+...+(2n))=\)

\(\lim_{n \rightarrow \infty}(n^2)-(n^2+n)=\lim_{n \rightarrow \infty}-n=-\infty\).

As you can see the sum diverges (tends to \(-\infty\)) and so is undefined. – Muhammad Shariq · 3 years, 6 months ago

Log in to reply

– Nanayaranaraknas Vahdam · 3 years, 2 months ago

The series of \(1-1+1-1+1\ldots\) is the infamous Grandi series, which yields the sums \(0, 1\) and \(\frac12\). However, for practical purposes, \(\frac12\) is the accepted value, as \(1\) and \(0\) can be yielded by just changing the parentheses.Log in to reply

– Muhammad Shariq · 3 years, 6 months ago

I'd like to add that the way your summing the series, it is even possible to get 1/8/, 1/16....and so on as the answer which is clearly absurd.Log in to reply

– Ashar Tafhim · 2 years, 1 month ago

yes it is possible . you should search rienmann series theorem which states that if a series is conditionally convergent it can always be re arranged to get any number.Log in to reply

– Nanayaranaraknas Vahdam · 3 years, 2 months ago

However, this series yields a finite sum, which is clearly defined, because of a unique property of infinity.Log in to reply