Why does the series $\sum_{n=1}^\infty\frac1n$ not converge? Can someone give a simple explanation as to why the harmonic series 


  $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$


doesn't converge, on the other hand it grows very slowly? 

I'd prefer an easily comprehensible explanation rather than a rigorous proof regularly found in undergraduate textbooks.
Let's group the terms as follows:

Group $1$ : $\displaystyle\frac11\qquad$           ($1$ term)

Group $2$ : $\displaystyle\frac12+\frac13\qquad$($2$ terms)

Group $3$ : $\displaystyle\frac14+\frac15+\frac16+\frac17\qquad$($4$ terms)

Group $4$ : $\displaystyle\frac18+\frac19+\cdots+\frac1{15}\qquad$   ($8$ terms)

$\quad\vdots$

In general, group $n$ contains $2^{n-1}$ terms.  But also, notice that the smallest element in group $n$ is larger than $\dfrac1{2^n}$.  For example all elements in group $2$ are larger than $\dfrac1{2^2}$.  So the sum of the terms in each group is larger than $2^{n-1} \cdot \dfrac1{2^n} =  \dfrac1{2}$.  Since there are infinitely many groups, and the sum in each group is larger than $\dfrac1{2}$, it follows that the total sum is infinite.  

This proof is often attributed to Nicole Oresme.
Given that the harmonic series of primes $ \sum_{p \, prime} \frac{1}{p}$ diverges, since 

$$ \sum_{n=1}^{\infty} \frac{1}{n} &gt; \sum_{p \, prime} \frac{1}{p}$$

it then follows that the harmonic series $\sum_{n=1}^{\infty} \frac{1}{n}$ also diverges.