Repeating Series as an Integral Approximation

So essentially, I have discovered an integral technique that uses series as overlapping Riemann sums to approximate an infinite integral.  Typical Riemann sums can approximate definite integrals, but if we use multiple geometric series which we can find the sum of and take the average of those series, we get an approximation pretty close to the actual value of the integral. Below I will demonstrate:

Essentially the series is each "staples" color and the value at each endpoint goes into the series' sum. As shown, the more staples you have, the more accurate the series approximation becomes. From my research, this is the first integral approximation technique of its kind, but if this is something that already exists, feel free to let me know.