If $1+2+3+4... = -1/12$
then, $(1+2+3+4...)*1/2$ should equal $-1/24$
But I find this strange since the second infinite is larger than the first because $1/2+2/2+3/2+4/2\dots$ contains all integers of the first group $(2/2,4/2,6/2,\dots)$ plus all the other fractions. But the sum of all its components is smaller.
I am seeing this wrong?
You can't really perform math on a divergent series and expect things to work logically.
For example, there are certain properties that a series might lose, most of these relating to the way we can manipulate the series and still get logical results.
Since the result of $-1/12$ was found in a way that made manipulating the series difficult, we cannot expect our result to be manipulatable in the common sense.