Can someone give a simple explanation as to why the harmonic series
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 : 11 (1 term)
Group 2 : 12+13(2 terms)
Group 3 : 14+15+16+17(4 terms)
Group 4 : 18+19+⋯+115 (8 terms)
In general, group n contains 2n−1 terms. But also, notice that the smallest element in group n is larger than 12n. For example all elements in group 2 are larger than 122. So the sum of the terms in each group is larger than 2n−1⋅12n=12. Since there are infinitely many groups, and the sum in each group is larger than 12, it follows that the total sum is infinite.
This proof is often attributed to Nicole Oresme.