Sum of geometric series

Given geometric series

Using the apllet of Sum of geometric series, the visual values of these elements will be displayed and the visual explanation for the sum of these can be given.

Using the applet of Parametric Equations

The button of "Clear" is for clearing the drawing.
The defalt values of a, r are 6, 0.7 respectively.
Where a is the first element and r is the ratio.
If you drag the point A1, a will be changed.
If you drag the line of "Y=rX", r will be changed.

First, click the button of "<=", then find the visual values of the elements.
Second, click the button of "Add", these elements will be added.
The button of "Add" will be changed that of "Sum".
Third, click the button of "Sum", then find the sum of these.
The button of "Sum" will be changed that of "Ans.".
If you click the button of "Ans.", the sum of these wil be showed.

The applet of sum of geometric series

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The sum of geometric series is the added value of pink bars in the picture on the left.
The sum, however, is not evident from these bars.

Now note the other pink bars.
The added value of these is (1-r) times the sum.
Besides, it can be given another expression by the picture.
That is, it is equal to.
Therefore the follwing equation can be find.

Now we reexpress as S.
So that

Let -1 < r < 1.
(the line of "Y=rX" is lower than that of "Y=X")
If , becomes 0.
So that