Math behind Keynesian Expenditure Multiplier

Question

Take a look at this page: http://wiki.ubc.ca/Keynesian_Multiplier
Why can you find out the sum of the geometric series just by dividing the mps by 1?

Answer 1

Consider the series $$ 1+r+r^2+r^3+\cdots=\sum_{k=0}^\infty r^k $$ If $|r|<1$, then this is a geometric series whose sum is the well-known $$ \sum_{k=0}^\infty r^k=\frac{1}{1-r}. $$ So if you have $MPC$ instead of $r$ and $MPS=1-MPC$, then this is $$ \sum_{k=0}^\infty MPC^k=\frac{1}{1-MPC}=\frac{1}{MPS}. $$