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?

This answer is hopefully to increase your intuition about summation. Imagine adding infinitely many (non negative, for simplicity) numbers together. Roughly speaking, if this addition adds up to a finite number, you say that the series (the terms you are summing together) converges, and if it doesn't, you say that it diverges. This is just a natural generalisation of finite addition to infinitely many terms. Lets look at this a little more closer.


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*For example, add the number $1$ to itself infinitely often. The result can not be anything finite, can it? So we say that the series $\sum_{n=1}^{\infty}1$ diverges.

*Consider then adding the numbers $1, \frac{1}{2}, \frac{1}{4},\frac{1}{8},...$ together. It turns out that they sum up to a finite number, which means that the series $\sum_{n=1}^{\infty}\frac{1}{2^{n}}$ converges. (In fact, the value of this sum equals $1$)

*What about if we add $1, \frac{1}{2}, \frac{1}{3},\frac{1}{4},...$ together? It turns out that these numbers don't add up to a finite number, in other words the series $\sum_{n=1}^{\infty}\frac{1}{n}$ diverges.
So it makes sense that in order for the series $\sum_{n=1}^{\infty}a_{n}$ for non-negative terms $a_{n}\in\mathbb{R}$ to converge, we must have $\lim_{n\to\infty}a_{n}= 0$. But is this enough? The answer is no, as we saw in the example above. But it turns out that if $a_{n}\to 0$ fast enough, then the given series $\sum_{n=1}^{\infty}a_{n}$ converges. For example, the sequence $(\frac{1}{2^{n}})_{n=1}^{\infty}$ converges to $0$ much more rapidly than the sequence $(\frac{1}{n})_{n=1}^{\infty}$.
To verify that the above example series add up to a finite number or not, you can use some known integral or convergence tests.
