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.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Consider that I have a path which start at point (0,0).

I need to find the ending point of the path using series.

The path look like this :

enter image description here

Any idea on how to start that ?

share|cite|improve this question
up vote 5 down vote accepted

The $x$ coordinate is $$8 \left(1 - \frac{3}{4} + \frac{3^2}{4^2} - \frac{3^3}{4^3} + \ldots \right)$$ and the $y$ coordinate is $$6 \left(1 - \frac{4}{5} + \frac{4^2}{5^2} - \frac{4^3}{5^3} + \ldots \right).$$ So you have to find the value of two geometric series.

share|cite|improve this answer
Alright, but could you please explain how you found they were the coordinate as this is mainly what I can't figure out? – MathLearner Jun 8 '14 at 7:34
@ProgrammerJeff The vertical parts do not contribute to the $x$ coordinate. So the $x$ coordinate is $8 - 6 + 9/2 - \ldots$. Similarly the horizontal parts do not contribute to the $y$ coordinate. – WimC Jun 8 '14 at 7:37
Not too sure to understand.. The answer would be $(\frac{32}{7},\frac{30}{9})$ ? – MathLearner Jun 8 '14 at 7:46
Even tho you did not answer, I tried another way and arrived to the same answer so you did well explain it I was just unconfident. Anyway, thanks. – MathLearner Jun 9 '14 at 23:18

The horizontal line co-ordinates are

$$8 , 6, \frac{9}{2}, \frac{27}{8}, ??? $$

These values give away the pattern,

$$\frac{9}{2}, \frac{27}{8}$$

WimC has explained the formula in his answer above, but even from this, I can deduce that the preceding values should be

$$ \frac{1}{0.125}, \frac{3}{0.5}, \frac{9}{2}, \frac{27}{8}$$

This would indeed keep with the first 2 horizontal values in your example

$$ \frac{1}{0.125} == 8, \frac{3}{0.5} == 6 $$

If I apply the same pattern to the inner part of the series, the inner most horizontal ... values should be.

$$ 8, 6, \frac{9}{2}, \frac{27}{8}, \frac{81}{32}, \frac{243}{256}$$

You can apply the same approach for the vertical value as well as they follow a similar pattern.

share|cite|improve this answer

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.