Examples

Geometric series - total sum and partial sums

The geometric series total sum can be calculated using a “shift technique” as follows: (1) Compare and :

(2) Subtract second line from first line, many cancellations:

(3) Solve to find :

Note: this calculation assumes that exists, i.e. that the series converges.

The geometric series partial sums can be calculated similarly, as follows:

(1) Compare and :

(2) Subtract second line from first line, many cancellations:

(3) Solve to find :

(4) The last formula is revealing in its own way. Here is what it means in terms of terms: