数列和 python

很多数学家一起去了一个酒吧，第一个要了一杯啤酒，第二个要了 1/2 杯啤酒， 第三个要了 1/4 杯啤酒， 第四个要了 1/8啤酒。 那个漂亮的bartender 翻了个白眼，给了他们两杯啤酒，说， 你们自己去分吧！ 

1+1/2+1/4+1/8+1/16... 1 +1/2+ 1/4 +1/8 +1/16 ... 1+1/2+1/4+1/8+1/16...

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RitaE / Pixabay[/caption]

This is a famous example of a Geometric series

KaTeX parse error: Expected 'EOF', got '·' at position 27: …+ 1/8 + 1/16 + ·̲ · ·

The fundamental idea is the base on the formula:

$ a^2 - b^2= (a-b)(a+b) $

$ 1-x^n= (1-x)(1+x+x^2 +x^3 +x^4…xn-1) $

For infinity series we have :
11−r=1+r+r2+r3... frac{1}{1-r}= 1+r+r^2 +r^3 ...1−r1​=1+r+r2+r3...

So the General Formula would be:
∑rk=r1−r sum r^k = frac{r}{1-r} ∑rk=1−rr​

Then we have the result to be 1+ 1 =2 Beers :

Now we can check if Python does a good job with computing the Series.

def SumofGeo(a,r,n):
    sum=0
    i=0
    while i 
Python Output:
Out[6]: 1.9375
Out[7]: 1.998046875
Out[8]: 2.0
As we can see when n=5, the geometric sum of (1/2)^n is 1.93, when n=10, the sum is 1.99. when n=100, python assumed it is 2. Remember theoretically, it is not 2 yet, the sum is 2 when n approaches infinity!
So the Hot BarTender is smart!!
Cheers and Happy Studying! ‍♀️
References:
https://en.wikipedia.org/wiki/Geometric_series
https://www.geeksforgeeks.org/program-sum-geometric-series/