Z transform of and exponentially decaying sequence

The series:          this converges for x < 1 : Both of these expressions are the Z transform of the  exponential decay sequence.  The first expression is easier to deal with because it is smaller and easier to work with.  The following diagram uses a decay sequence with

The filter takes what ever input it is fed and every interval multiplies it by 0.75.   To see the filters time response you can thus feed in a single ping.  This results in the filter output tracing out a special response.  This  is called the impulse response. It is the same as the filter plot above.

The exponential decay is maximum entropy.  That is to say this is how concentrated things soak out into the rest of the world as they become more dilute.