Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In the drawing on Geometric Series Proofs: An Annotated Bibliography, using Fig. 4 on the page, Please explain why it is said that "ON is the limit of the sum 1+x+..."? Thank you.

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

share|improve this question
add comment

2 Answers

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 $ON$.

share|improve this answer
add comment

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 $1 + x + x^2 + \dots$

share|improve this answer
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.