Why does the harmonic series diverge but the p-harmonic series converge I am struggling understanding intuitively why the harmonic series diverges but the p-harmonic series converges. I know there are methods and applications to prove convergence, but I am only having trouble understanding intuitively why it is. I know I must never trust my intuition, but this is hard for me to grasp. In both cases, the terms of the series are getting smaller, hence are approaching zero, but they both result in different answers.
$$\sum_{n=1}^{\infty}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots = \text{diverges}$$   $$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^2}=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots =\text{converges}$$

Intuitively the main argument why the harmonic series diverge is that $\forall k \sum_{n=k}^{n=2k}\frac{1}{n}>k\frac{1}{2k}=\frac{1}{2}$ since smallest element is $\frac{1}{2k}$ and there are k elements in the interval $[k;2k]$. So the harmonic sum for any finite interval $[k;2k]$ is > 0.5.
So if you split the infinite interval over which you make the summation into intervals $[1;k][k;2k][2k;4k],...,[2^nk;2^{n+1}k],...$ each interval has sum higher then 0.5 and since there are infinite number of this intervals to cover the whole $\mathbb{Z}$ it diverges.
Basically they get smaller and smaller, but not fast enough to converge to a limit. 
The p-harmonic on the other hand because of the square in the denominator can not have this "ability" and converge, aka they get smaller faster enough.
To try to explain it on intuitive level better thing like this: you are adding infinite number of the sequence, so in order to converge to a limit L they have to get smaller with some "speed". Now even if they get close to 0 if the speed at which they grow smaller is not high enough then they still will be too much stuff added and will never converge.
