Examples

Geometric series - total sum and partial sums

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

$$\begin{array}{cc}S=& a_0+a_{0}r+a_0{r}^2+a_0{r}^3+\cdots \\ {\gg \gg }^{\times r}rS=& a_{0}r+a_0{r}^2+a_0{r}^3+a_0{r}^4+\cdots \end{array}$$

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

$$\begin{array}{cc}S& =a_0+a_{0}r+a_0{r}^2+a_0{r}^3+\cdots \\ -(rS& =a_{0}r+a_0{r}^2+a_0{r}^3+a_0{r}^4+\cdots )\\ \text{—}\text{—}\text{—}& \text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\\ S-rS& =a_0\end{array}$$

(3) Solve to find $S$:

$$S=\frac{a_0}{1-r}$$

Assumes convergence!

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

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

(1) Compare $S$ and $rS$:

$$\begin{array}{cc}{S}_N=& a_0+a_{0}r+a_0{r}^2+\cdots +a_0{r}^N\\ {\gg \gg }^{\times r}r{S}_N=& a_{0}r+a_0{r}^2+\cdots +a_0{r}^N+a_0{r}^{N+1}\end{array}$$

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

$$\begin{array}{cc}{S}_N=& a_0+a_{0}r+a_0{r}^2+\cdots +a_0{r}^N\\ -(r{S}_N=& a_{0}r+a_0{r}^2+\cdots +a_0{r}^N+a_0{r}^{N+1})\\ \text{—}\text{—}\text{—}& \text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\text{—}\\ S_N-r{S}_N& =a_0-a_0{r}^{N+1}\end{array}$$

(3) Solve to find $S_N$:

$$\begin{array}{c}{S}_N=a_0\frac{1-r^{N+1}}{1-r}\\ \\ =\frac{a_0}{1-r}-\frac{a_0}{1-r}{r}^{N+1}=S-S{r}^{N+1}\end{array}$$

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

$$\begin{array}{c}{a}_0+a_{0}r+\cdots +a_0{r}^N=\\ \\ & \begin{array}{cc}{a}_0+a_{0}r+a_0{r}^2+\cdots & \\ & -(a_0{r}^{N+1}+a_0{r}^{N+2}+\cdots )\end{array}\end{array}$$