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 :)

#mathshorts​ #mathvideo​ #math​ #calculus #mtbos​ #manim​ #animation​ #theorem​ #pww​ #proofwithoutwords​ #visualproof​ #proof​ #iteachmath #geometricsums #series #infinitesums #infiniteseries #geometric #geometricseries #rectangle #silverratio #goldenratio

To learn more about animating with manim, check out:

[Chikowski]

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.

[heavysaur]

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.

[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**.

[strikerstone]

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

[birajshankarsingh]

Where were you....when I was in 11th and 12th std😢