Here’s a nice little exercise:

1. Find the first 20 or so digits of 1/9899.

2. Marvel at the beauty.

3. Explain.

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Here’s a nice little exercise:

1. Find the first 20 or so digits of 1/9899.

2. Marvel at the beauty.

3. Explain.

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I don’t want to give this away yet, but I suggest repeating steps 1 and 2 for 1/998999, 1/99989999, 1/9999899999, etc.

Now, I knew about the 1/81, 1/9801, 1/998001, … family, but this is even more amazing!

Wow wow beautiful exercise!! so.. generating functions allow us to create any (¿? at least coming from a recursive sequence) “special” number we want! Let $a_0$ and $a_1$ two integers and construct a sequence satisfying the recurrence formula $a_{n+1}=a_{n}+a_{n-1}$. This sequence has the generating function $F(x)=\frac{a_0+(a_1-a_0)x}{1-x-x^2}$. Show time! Take $a_0=a_1=1$ and see which numbers are $F(1/10)$, $F(1/100)$… (well up to a decimal scaling!)

501/499499 is pretty cool too