Odd Series Behavior: Exploring 1 + -1 + 1 + -1

[bowma166]

Today, my professor said something like "The series 1 + -1 + 1 + -1 and so on is defined to be one half... but let's not go into that." and then didn't feel like explaining when people asked him why. I have no idea why that would be true...

It seems like a similar case might be

$$\int_{0}^{\infty}\sin x\,\textrm{d}x$$

but that isn't defined to be one half or zero or anything at all.

So why oh why is this true?

$$\sum_{n=0}^{\infty}\left(-1\right)^{n}=\frac{1}{2}$$

 

[sutupidmath]

So why oh why is this true?

$$\sum_{n=0}^{\infty}\left(-1\right)^{n}=\frac{1}{2}$$

THis is true...because it is NOT true!

the alternating series
$$\sum_{n=0}^{\infty}\left(-1\right)^{n}$$

does not converge at all.
One will get either 1 or 0 as the final sum, depending on how you group the terms. So, by the definition of what we mean with convergence it does not converge.

BUT, i have heard of some sort of Eulers method, or Ramaujan summation, or some different kind of summation, and that might be true, but i have no knowledge whatsoever of those summations.

So, this is not true, if the summation is the common one, i don't know about the others. But people here will enlighten you, just wait until this thread catches the eyes of the right people...

 

[qntty]

Although the series diverges, there are methods of assigning a value which yeilds 1/2

 

[yyat]

Another reason why one might want the sum to be 1/2 is that

$$\sum_{n=0}^{\infty}x^n=\frac{1}{1-x}$$

for $$-1<x<1$$. If you put in $$x=-1$$, then the series on the left diverges, but on the right you get 1/2. This is basically the idea behind "Abel summation", which is a weaker form of Cesaro's method.

 

[Santa1]

I've usually seen Ramanujan summation being used in cases like that.

Equivalently you can write it in terms of the zeta function, by
$$\eta(s)=\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n^s} = (1-2^{1-s})\zeta(s)$$
so that,

$$\sum_{n=0}^\infty (-1)^{n} = \eta(0) = -\zeta(0) = \frac{1}{2}$$

If you have seen the derivation of the functional equation (which is as far as I have understood is Ramanjuan summation) for the zeta function this sort of "makes sense".