Why do some series converge and others diverge? Why do some series converge and others diverge; what is the intuition behind this? For example, why does the harmonic series diverge, but the series concerning the Basel Problem converges?
To elaborate, it seems that if you add an infinite number of terms together, the sum should be infinite.  Rather, some sums with an infinite number of terms do not add to infinity.  Why is it that adding an infinite number of terms sometimes results in an answer that is finite?  Why would the series of a partial sum get arbitrarily close to a particular value rather than just diverge?

Here's an intuitive answer
When a series converge it's because that the series goes towards a target, its limit. Likewise a diverging series has no target, it either jumps around in circles or goes to an infinite value. The harmonic series diverges because, even though it increases by smaller and smaller amounts, it will still never actually end at a target, basically for any value n there is some iteration of the harmonic series which has a larger value than that. It just flies away.
