Show that the series $$\sum_{k=1}^{\infty} (-1)^n$$ is divergent. Explain what is wrong with the following reasoning:
Guido Ubaldus made something out of nothing:
$$0 = (1 - 1) + (1 - 1) + (1 - 1) + \dots$$
$$= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + \dots = 1$$
I do not understand is why he was allowed to move the round brackets? Do mathematicians get the right, to do this or is it something he just did for this experiment? (What he did is against algebraic manipulation correct?)

Question

Show that the series $$\sum_{k=1}^{\infty} (-1)^n$$ is divergent. Explain what is wrong with the following reasoning:
Guido Ubaldus made something out of nothing:
$$0 = (1 - 1) + (1 - 1) + (1 - 1) + \dots$$ 
$$= 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + \dots = 1$$
I do not understand is why he was allowed to move the round brackets? Do mathematicians get the right, to do this or is it something he just did for this experiment? (What he did is against algebraic manipulation correct?)

experimentx · Answer

first you should look into definition of convergence ... then you will know it's divergent or convergent.

anonymous · Answer

I used geometric series test with r = -1 to determine if the series is divergent.
Which it is, meaning there is no limit correct? So the sum or this series cannot be 0 or 1.

Although that is not exactly what I am trying to ask, I want to ask why the philosopher and mathematician Guido Ubaldus was allowed to break algebraic manipulation? (He moved the round brackets without regard to algebraic manipulation.) Did he do this to experiment a possible idea, or is there a valid reason for it?

experimentx · Answer

use this definition http://en.wikipedia.org/wiki/Convergent_series $$ |S - l| < \epsilon $$

experimentx · Answer

apparently it does not have one particular 'l', so it does not converge.