# Math behind Keynesian Expenditure Multiplier

Why can you find out the sum of the geometric series just by dividing the mps by 1?

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I see no MPS, only MPC. Is that what you mean? And what is divided by 1? I can't see where that is. Can you be a little bit more specific? – hejseb Nov 19 '13 at 16:41
ah sorry lol, MPS equals to 1 - MPC – Mouse Hello Nov 19 '13 at 16:49
@hejseb is there any other thing ambiguous in my question? – Mouse Hello Nov 19 '13 at 16:56
Oh, I see. I should probably have figured that out myself... See my answer below. Hope it clears some confusion up. – hejseb Nov 19 '13 at 16:59

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 $$\sum_{k=0}^\infty MPC^k=\frac{1}{1-MPC}=\frac{1}{MPS}.$$