Background Story :
There were a couple of mathematicians went to a bar, the first guy ordered one beer, the second guy asked for 1/2 of a beer, the third guy asked for 1/4 of a beer, the fourth ordered 1/8 of a beer, and so on… The hot bartender rolled her eyes, poured two beers, and says, “Here, you guys work it out ! “ 😍
Question: What is the Summation of
1 +1/2+ 1/4 +1/8 +1/16 ...
This is a famous example of a Geometric series
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....x^n-1)
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)
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! 🙇♀️