*Originally posted by adam warlock*

**To an infinite sum to converge, that is to have a sum, its general term must tend to zero. This a necessary but not sufficent condition. A series can be written like this S u_n
**

For instance S(1/2)^n where S denotes a *sum* from zero to infinity. This is a convergent series whose value is 2 and the general term is (1/2)^n

in the 1-1+1-1... series ...[text shortened]... whose general term goes to zero but never the less diverges. S(1/n) being one simple example.

Of course you're right. I've never doubted that. My math professor says it, the book says it, I say it myself.

But just for the fun of it...

Suppose you have a series S with terms of this looks:

S(n) = -1 if n is prime and +1 otherwise.

What is the sum of S when n goes from 1 to infinity?

One way to see it is to count the primes and compare this number with the number of the other numbers.

(The sum of S(n) when n goes from 1 to 20 is -1-1-1+1-1+1-1+1+1+1-1+1-1+1+1+1-1+1-1+1=+2. The higher n the higher sum.)

Surely there are more number that is not primes than the number of primes, right?

Wrong! There are exactly the same number primes as there are non-primes.

So we are actually led to believe that the sum of S = 0 !!!

But this is just for the fun of it. Somewhere this reasoning is wrong. One can't do like that.

The limit of S when n approaches infinity is not defined at all, of the same reasons you wrote.