A Physical Approach to The Basel Problem by Jinkai Li (12JAR) , Bryan Lu (12JLG)
1. Introduction
The Basel problem states that,
1 1 2
+ 1 2 2
+ 1 3 2
+ ⋯+ 1 2
+ ⋯ = ？
Actually, this is equal to π 2 6
. It is amazing and unbelievable, isn’t it? How the sum of inverse square could
add up to an irrational number which is related to a circle? Many people do know the way to tackle it by
Puller’s method. However, it is difficult for upper school students to understand. Today, we are going to
show you an interesting understandable physical approach to this problem.
A naive way of attacking this problem is to add up as many terms as you can and see what you get. The
problem with the Basel sum, however, is that it converges very slowly. Add up the first 1000 Terms only
gives you a result that is correct in the first two decimal places, but differs the third. 【 1 】
We want to find more accurate result. Hence, we provide a geometric method.
2. Proof that the value is convergent
1 (+1)
, ⋯ , a n
, ⋯ , which a n
We can construct an infinite sequence a 1 , a 2 , a 3
=
S= 1 2
+ 1 6
+ 1
+ 1
+ 1
+ ⋯
12
20
30
Assume, A= 1 1 2
+ 1 2 2
+ 1 3 2
+ ⋯+ 1 2
+ ⋯
1 + S= 1 + 1 2
+ 1 6
+ 1
+ 1
+ 1
+ ⋯
12
20
30
1 2
> 1 2 2
1 6
> 1 3 2
1 12
> 1 4 2
1 (+1)
1 (n+1) 2
⋯
>
Therefore 1+ s >A
1 + s=1+1− 1 2
+ 1 2
− 1 3
+ 1 3
+ ⋯ < 2
Hence A< 2
We conclude that it is converge(less than 2)
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