Convergent or Divergent ?

[stanek]

Please forgive my ignorance here. Assume we have a series

S(n)=1/1^2 - 1/2^2 - 1/3^2 + 1/4^2 - 1/5^2 + 1/6^2 - 1/7^2 + 1/8^2 + 1/9^2 + 1/10^2 ... +/- 1/n^2

where we subtract 1/n^2 if the number is prime and add 1/n^2 if it is not prime.

Can this even be considered a series when it contains an 'If' condition? Is there another way to phrase this so that it works the same way?

If yes, is this a convergent or divergent series? On one hand it seems divergent because as the series grows, the ratio of non prime numbers to prime numbers grows, thus making its value increase. But for some reason I have it in my head that for something to be divergent it means that as

n-> oo(infinity) then S(n)->oo

which doesn't seem to be the case here because there are an infinite number of primes to subtract from the series.

I'm thinking that this isn't really a series but I'm not positive, thats why I am asking.

[borisp]

This is a series, and it is convergent because it converges absolutely; that is, the series obtained by simply adding 1/n^2 every time converges.

To show that 1/n^2 converges absolutely, you integrate the function 1/x^2 from 1 to infinity and observe that the integral is finite.

To define a series is really to define its sequence of partial sums. Recursion is a perfectly good method for defining a sequence, and so yes, this is a series, since it is defining the partial sums recursively.

[jay_shark]

S(n) can be written as

S(n) = 1/1^2 +1/2^2 +1/3^3 +.... -P(n)

where p(n) = 2[(1/2^2)+1/3^2 +1/5^2 +...]

Notice that S(n)+P(n) = pi^2/6 which is a well known result.
If P(n) converges , then S(n) must certainly converge . It should be obvious that P(n) is less than S(n) and positive , which should imply that S(n) is convergent .

Here is a simple way to show that S(n)+ P(n) converges . I will show that this number never exceeds 2 .

1/1^2 +1/2^2 +1/3^2 +.....
=1+ (1/2^2+1/3^2) + (1/4^2 +1/5^2 +...1/7^2) + (1/8^2 +...1/15^2) +....
< 1 + 1/2 +1/4 + 1/8 + ... = 2 ; which is a geometric series .

So I've showed that this series definitely has an upper bound . You don't even need to show that this series converges to pi^2/6 . As long as you show that it's strictly less than some finite number and that P(n) is less than some finite number . The difference between two finite numbers is finite .