Background Story :
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！ 😍
Question: What is the Summation of
RitaE / Pixabay[/caption]
Solution:
This is a famous example of a Geometric series
The fundamental idea is the base on the formula:
For infinity series we have :
So the General Formula would be:
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 <n:
sum=sum+a
a=a*r
i=i+1
return sum
SumofGeo(1,1/2,5)
SumofGeo(1,1/2,10)
SumofGeo(1,1/2,100)
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/
