Geometric Series: Deriving the Sum using an Ingenious Method (and via Similar Triangles)
In this video, I go over the geometric series, which is the sum a + a r + a r^2 + ..., and show that if the absolute value of the common ratio r is less than 1, then the sum is just the first term divided by one minus the common ratio. I first derive this formula by a very simple yet genius method of multiplying the infinite series by r, and then subtracting the resulting series and rearranging to solve for the sum. I also illustrate this geometrically via similar triangles. For all other values of r, the series is divergent, hence does not exist.
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