Convergent or Divergent ?
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.
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