The limit of the infinite series 2/5 + (2/5)2 + (2/5)3 + (2/5)4 + ... equals the area within a triangle

by Don

Don shows below that the limit of the infinite series 2/5 + (2/5)2 + (2/5)3 + (2/5)4 + ... = the area within the orange triangle. Note that the terms of the series alternate being tall rectangles (the odd powers) and squares (the even powers). Don connected the orange diagonal line from the point (0,1) through the upper right corners of the squares. He found the slope of this line to be -3/4 and the y-intercept 1; then he found the x-intercept to be 4/3.

                                 

Notice that the piece above the diagonal orange line in the the rectangles, equals the area below the orange line above the square to

 its right. Notice also that the length of the base of the orange triangle =

 2/5 + 2/5 + (2/5)2 + (2/5)2 + (2/5)3 + (2/5)3 +(2/5)4 +(2/5)4  ... = 2* [2/5 +

 (2/5)2 + (2/5)3 + (2/5)4 +  ...]. The area within the orange triangle =

 (1/2)(base x height) = (1/2)(2x [2/5 + (2/5)2 + (2/5)3 + (2/5)4 +  ...])x1= 2/5 +

 (2/5)2 + (2/5)3 + (2/5)4 + ... , the sum of the infinite series whose limit is

 2/3. This is also 1/2x4/3x1= 2/3 and also 2/(5-2) = 2/3. This last one

 comes from doing lots of series and students generalizing A/B +

 (A/B)2 + (A/B)3 + (A/B)4 +  ... -> A/(B-A).

Also see Lori's (and Don, a little), applet in GeoGebra, at Area of triangle= limit of infinite series

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