>>12129189Of course it matters. Think about it this way.
What is the sum
Well, if we take a line of length 1, cut it in half, then cut one of the halves in half again, then one of the quarters in half again, and so on, we get lines with this sequence of lengths. So we know the sum is 1.
Now, how about the sum
Now we're taking an infinite ray and splitting it up. We split off a line of length 1, then another line of length 1 which we cut into halves, then another line of length 1 which we cut into quarters, then another line of length 1 which we cut into eighths, and so on.
The resulting sum is obviously infinite because we cut up the whole ray, and the ray was infinitely long.
Note that the first sequence drops very quickly while the second sequence gets slower and slower the further you go along.
The harmonic series is just like the second sequence, you can check that you can split up the sequence into an infinite bunch of little blocks, each taking up at least 1 length. Then all these blocks together make infinity.
On the other hand, the factorial sequence is like the first one. The further terms fall so quickly that if you lay them end to end they stay in a bounded area.
