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?
Best Answer
A series converges if the partial sums get arbitrarily close to a particular value. This value is known as the sum of the series. For instance, for the series $$\sum_{n=0}^\infty 2^{-n},$$ the sum of the first $m$ terms is $s_m = 2-2^{-m+1}$ (you can figure this out using the fact $1+x+x^2+\cdots+x^n = (x^{n+1}-1)/(x-1)$). Since $s_m$ tends to $2$ in the limit as $m$ gets large, the sum is $2$. In this case we can represent the partial sums as a formula and think of it as a limit. If you need a visualization, consider the following image from this thread.
It turns out that if $\sum_{n=0}^\infty a_n$ converges, we must have $a_n \to 0$ as $n \to \infty$. But just because $a_n$ goes to 0 doesn't mean the sum converges.
For instance, the partial sums of $\sum_{n=0}^\infty \frac{1}{n}$ go to infinity even though $1/n \to 0$ as $n \to \infty$. Look up the integral test or questions about the divergence of the harmonic series to learn why.
On the other hand, the series $\sum_{n=0}^\infty \frac{1}{n^2}$ does converge, to $\pi^2/6$, in fact. We can show that it converges using various theorems, one of them includes the integral test. To find the value of the sum requires more work.
So at the end of the day, we have to use specific tools to show specific series either converge or diverge. There's no complete algorithm for figuring this out that is taught (or even exists as far as I know).