After watching the infinity elephants video http://www.youtube.com/watch?v=DK5Z709J2eo and seeing how a geometric series could be represented by drawing a circle between a pair of lines, then the largest circle that would fit in the gap etc. I wondered about using curves instead of lines to find other types of series. For example, circles with radii in arithmetic progression inside a parabola. What kind of curve would go around a line of circles with radii $1,\frac{1}{2^n},\frac{1}{3^n}...$, for example?
summing series using circles inside curves
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0I would be extremely happy if I could draw circles with that precision free handedly :-). (The same applies to elephants, but with smaller amount of longing). I think that the author has quite some insight into 2d (and 3d) geometry, which -- as far as your question is concerned -- would make this probably a prerequisite. – 2011-12-25
1 Answers
For a given series $(a_1, a_2, a_3, \ldots)$, we may draw the first circle centered at $\left(\frac{a_1}2,0\right)$ with radius $\frac{a_1}2$, the second at $\left(a_1+\frac{a_2}2,0\right)$ with radius $\frac{a_2}2$, and so on. For simplicity, I'm going to let the curve pass through the highest points of the circles, rather than tangent to them. Then we have a sequence of points $\begin{align} x_n &= \sum_{i=1}^{n-1}a_i+\frac{a_n}2,\\ y_n &= \frac{a_n}2, \end{align}$ which may have a simple curve passing through them. For example, for the geometric series $a_n = \alpha r^n$, we have $x_n = \alpha r\frac{1-(r^{n-1}+r^n)/2}{1-r}$ and $y_n = \frac{\alpha r^n}2$, which does indeed yield a linear equation, $(1+r)y_n = \alpha r - (1-r)x_n$. If we have your sequence $a_n = n^{-s}$ instead, well... a closed form is out of my reach, but it might have to involve the Riemann zeta function.