I duck duck go'ed the money multiplier effect myth. From mises.org :
They argue that in the present US monetary system there is no such thing as a money multiplier since banks make loans first and worry about reserves later. Moreover, it is argued, within the present lagged reserve requirement framework it is pointless for the Fed to pump reserves, for these reserves cannot be used by the banks since required reserves are only calculated on the past 30 day's deposits.
This is the type of misdirection that keeps the public ignorant in understanding that any money multiplication effect, as it occurs from lending without usury, makes no difference if the reserve requirement is met a priori or a posteriori.
In other words, the effective mathematical difference is zilch, if a loan is made from deposits (as in my example) or from bankers making rational banking decisions as to creating loans and then loaning from a Central Bank to meet reserve requirements.
The money multiplier effect still exists in either scenario and is based on the mathematically sound principles of convergent series whose elements in the sequence are finite partial sums.
Now even if this is not a perfect infinite geometric ratio in the real world, the implications of how this model can be easily seen with how money is created, regardless if loans are done via rational business decisions vs. loaning of new deposits, are readily seen by any casual observer.
Well, it can't be a fringe theory, it comes from the mises institute...
The mises article is actually supporting the MM effect as a real thing... I posted a list of links that included that one for the sake of fairness, as its the best one that actually supports the money multiplier theory.
The Mises article quote is a characterization of the claims of what those who advocate the myth of MM ... my apologies for not making that abundantly clear.
Regardless, the fact of the matter is that you did not adequately address my concerns of money multiplication, due to a posteriori or a priori meeting of reserve requirements, and how this relates to a limiting value in an infinite sequence, in which there are an infinite elements of the sequence, each of which is a finite partial sum.