Odd series behavior?

1. Mar 13, 2009

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?

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

2. Mar 13, 2009

sutupidmath

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....

3. Mar 14, 2009

qntty

4. Mar 14, 2009

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.

5. Mar 14, 2009

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".