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The following image is from Geometric Series Proofs: An Annotated Bibliography.

enter image description here

Please explain why it is said that:

"$ON$ is the limit of the sum $1+x+\dots$."

Thank you.

Edit: I guess what through me off was the word "limit"!

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If anyone is interested, several years ago I managed to trace this approach back to Milutin Milankovitch [Milankovitsch], Eine graphische darstellung der geometrischen progressionen, Zeitschrift für mathematischen und naturwissenschaftlichen 40 (1909), 329. I do not know whether this idea had been published earlier than 1909. In the 1950s Constantin P. Orloff published several follow-up papers on this in a Yugoslavian journal. – Dave L. Renfro Jun 6 '14 at 14:08
@DaveL.Renfro I appreciate the historical detail. – NoChance Jun 9 '14 at 20:38
up vote 2 down vote accepted

Notice that the horizontal lines of length $1,x,x^2,x^3,\ldots$ union to form a line of length $ON$.

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There are two ways to travel the horizontal distance from $O$ to $N$. The easiest way is to go via the line $ON$. Alternatively, you could follow the staircase, only counting the movement in the horizontal direction. The first stair has horizontal movement $1$, the second has horizontal movement $x$, and so on. However, once you've arrived at $N$, it shouldn't matter which way you went, the distances are the same. Therefore (the length of the line segment) $ON$ is the same as the limit of the sum $1 + x + x^2 + \dots$

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Why are the length of the line segments AB, CD, EF, etc, $x$, $x^2$, $x^3$, etc, respectively? I perceive that multiplying each base with the slope $x$ renders these results, but I do not understand how.

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