**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 ...

**Solution:**

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)

**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/