Sum Geometric Series with Ratio 1/2 in Rectangles (visual proofs)

In this video, we show an animated version of a recent proof without words that finds the sum of the geometric series with first term 1/root(2)+1/2 and ratio 1/2 using the unit square. Then we show how to find other geometric series sums with ratio 1/2 using different sizes of rectangles. Careful, the final bonus question might require a different technique than rotating :)
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Also, check out my playlist on geometric sums/series: • Geometric Sums
This animation is based on a proof by Ángel Plaza in the June 2024 issue of Mathematics Magazine (doi.org/10.1080/0025570X.2024... page 328).
#mathshorts​ #mathvideo​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #rectangle #silverratio #goldenratio
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[@Chikowski101]

this is the most beautiful mathematical video I have ever seen !!

[@kmjohnny]

This is really clever. For the golden ratio, this is the sum of 2 sides of the triangle that greek mathematicians used to find it, and called it extreme and mean ratio. It's very interesting to see how closely the silver ratio is related.

[@heavysaur149]

Interesting but it doesn't work for any generic a and b. The rotation process can overlap when b is less than 2a. It gives a perfect line only when b = 2a or when b = a if you rotate the other way around (shown with the example of a square of size length 1).
We can prove that by doing the sum:
We have for any generic a and b this sum:
(a/2 + d/2) + (a/4 + d/4) + .... = (a+d)/2 + (a+d)/4 + ... = (a+d) * (1/2 + 1/4 + 1/8 + ...)
Which is this sum:
(a + d) * Sum{for all positive integer k} (1/2^k)
But the sum of the geometric series is equal to 1 (we can know that with a visualization of a square being filled by 1/2, then 1/4, then 1/8... to reach the full 1 square.
(a + d) * Sum{for all positive integer k} (1/2^k) = a + d
If the rotation process succeeds it will be equal to b/2 + d so:
a + d = b/2 + d
a = b/2
b = 2a
We find that if the rotation process is successful then the rectangle needs to have a ration of 2.
Now if the rotation the other way works well, then it will be equal to b + d so:
a + d = b + d
a = b
b = a
If the rotation process is successful the other way around then the rectangle is a square.

[@MathVisualProofs]

Nice work! You can see them overlap in the fast animation near the end so I agree the same exact argument won’t work :) But can you see some other way to find the zig zag sum (instead of rotating, can we shift the vertical line segments parallels to main diagonal and get something interesting?

[@heavysaur149]

Yeah, seen in the calculation we find that the zig zag lines just gives a + d at the end. We are essentially just doing 1/2 + 1/4 + 1/8 + … to infinity with a and d at the same time which gives a cool looking zig zag.
Which can be seen if we put all the vertical lines on top of each other.
It in a sense gives a proof to why 1/2 + 1/4 + 1/8 + … = 1

[@natashok4346]

d+a

[@mauschen_gaming]

Answer:
1. Measure the lengths of the lines.
d/2 + a/2 + d/4 + a/4 + ...
2. Group a and d together.
(a + d)/2 + (a + d)/4 + ...
3. Convert to geometric series.
∞
Σ (a + d)/(2^k)
k=0
4. Use S = a_1/(1 - r) and plug in the values.
S = (a + d)/2 / (1 - 1/2)
S = (a + d)/2 / 1/2
**S = a + d**
Alternatively you can rotate the horizontal lines by 90 degrees and stick it on the a side and bring up the diagnals to the d side; this will fill all of the a and d lines also giving us **a + d**.

[@MathVisualProofs]

Nice! It’s too bad the vertical segments in the general diagram overlap when they rotate but you can shift them to the small side :)

[@mauschen_gaming]

@@MathVisualProofs yeah true when i read my comment again i just realized that you can just stack the vertical lines on the a side

[@MathVisualProofs]

@@mauschen_gaming nice to have lots of alternatives though!

[@mauschen_gaming]

Thanks

[@strikerstone]

Hey i was wondering can we you prove basel problem using visual methods