<h1><center>Basel Problem</center></h1>

Hi, Steemians, how are you guys doing? Hope **All is Well**. How many of you know the full form of STEM in steemstem? Its science, technology, engineering and mathematics. Well, we can see lots of posts related to physical/life science, technology, and engineering in steemstem. But, there are very few posts related to mathematics. So, if community members are interested, I will post many interesting problems in the field of mathematics.

In this very post, I want to discuss one of the important problems in mathematics, **Basel problem**. Its result and approach in solving this problem may interest many individuals in mathematics.

<div class="center"><center><img src="http://i3.cpcache.com/product/1601949093/basel_problem_solved_by_euler_mathematics_mugs.jpg?side=Back&width=750&height=750&Filters=%5B%7B%22name%22%3A%22background%22%2C%22value%22%3A%22F2F2F2%22%2C%22sequence%22%3A2%7D%5D" /><br/><em><a href="http://i3.cpcache.com/product/1601949093/basel_problem_solved_by_euler_mathematics_mugs.jpg?side=Back&width=750&height=750&Filters=%5B%7B%22name%22%3A%22background%22%2C%22value%22%3A%22F2F2F2%22%2C%22sequence%22%3A2%7D%5D">Source</a></em></center></div>


**Statement**

http://quicklatex.com/cache3/9d/ql_1f47776cd730cab1dcd96ad71ccad89d_l3.png

What is the closed form of this infinite sum ?

The closed form of a function is a function written without infinite sum. Since above problem is not a function but just a sum, closed-form means an exact value of the infinite sum.


**History**

Basel is a Swiss city where two Bernoulli brothers, Jakob Bernoulli(1687−1705) and Johann Bernoulli(1705−1748) served as professors successively. It is also the hometown Leonard Euler, who solved this problem. Basel Problem was after this city. It was first proposed by Pietro Mengoli in 1644. The first thing that comes to our mind when we see this problem is the harmonic sum. Both the Bernoulli brothers including Mengoli proved the divergence of the harmonic sum. However, Basel problem was convergent. So, Jakob Bernoulli had stated this problem and had asked anyone to get and prove that result. The sum was astonishing and later had implications with Riemann Zeta function.

**Discussion**

We know that harmonic series is not converging. Not converging means, the limit of the partial sums is not finite or do not converge to the same value. If every term of Basel series was greater than that in harmonic series, it would be self-evident that it is converging. But, it is not so.  The later terms in Basel sum are the square of the harmonic sum. So, the term is lesser than that of the harmonic sum. 

http://quicklatex.com/cache3/da/ql_7b9a3cc02f00d7711767322e86c84bda_l3.png

http://quicklatex.com/cache3/0e/ql_39089d620419ed34e9ddf86ecc3e830e_l3.png

Harmonic series diverges very slowly. So, let us hope that the series with every term much smaller to be converging. To find which number it converges to, let us find its partial sum. 

first term = 1
 
Sum of first two terms =  http://quicklatex.com/cache3/e2/ql_43208734e39680f3482b56fe6dc50fe2_l3.png =  1.25 

Similarly,

Sum up to| Sum
-|-
10| 1.5497677...
100| 1.6349839...
1000|  1.6439345...
10000| 1.6448340...

It can be observed that sum is somewhere around 1.644. This is merely an approximation. Maybe physicist or engineers would be happy enough from this approximation but not the mathematicians. We need closed form here, not an approximation. 


**The Result**

Here I disclose the result.

http://quicklatex.com/cache3/2c/ql_1a1a5392e7a6cad8023511c8897daa2c_l3.png

The result is astonishing.  http://quicklatex.com/cache3/e3/ql_67ff1b09f216c113ec2273ac26c062e3_l3.png is the ratio of circumference to the diameter of the circle. How can it be related to a random infinite sum not related to circle in any way? When we say equation or result is beautiful, it doesn't necessary mean that the equation and te result have profound application. In the way, the result above is totally unexpecting and relating two branches of mathematics together, without any application outside the paper, it is beautiful. Before giving you formal proof, let me give you other astonishing results.

http://quicklatex.com/cache3/7f/ql_09837b85b04169eefaf4ad093a4ce37f_l3.png


**The Proof**

With these results, I think you are more interested in the proof of this result now. It was first proved by Leonard Euler in 1735. Well, it can be proved by using other methods like Fourier series but the easiest would be proof using Euler's method. 

Probably, you are familiar with sine function in trigonometry. If 'x' is the reference angle then the ratio of perpendicular to the hypotenuse of a right-angled triangle is sin(x). It can be written in infinite series like;

http://quicklatex.com/cache3/0f/ql_03f1cb0282bffd1b47278131fb3a5b0f_l3.png

Here, sin(x) is the closed form of infinite series on the right-hand side. It can be shown using Taylor series. Let us just accept this for now.  

i. Replace x by http://quicklatex.com/cache3/c9/ql_d3692317aca2403a9a00eec665d55dc9_l3.png:

http://quicklatex.com/cache3/85/ql_e13cb07d828877fcb6c1cbb614d96385_l3.png

ii. Divide result by http://quicklatex.com/cache3/c9/ql_d3692317aca2403a9a00eec665d55dc9_l3.png:

http://quicklatex.com/cache3/71/ql_bf75e467ea1dff15a841b208424eef71_l3.png

iii. Rearranged:

http://quicklatex.com/cache3/9e/ql_f9f7db5d815a33872f37aa7adb8f779e_l3.png

Let this be equation (a)

iv. New relation

http://quicklatex.com/cache3/0a/ql_395001cfb6f50bbbcc83b8301ab8fd0a_l3.png

I won't prove rigorously but I will explain. 

We know the zeros of the function in LHS occur when x = 1,2,3,4 ...
In the RHS also zeros occur for the same values. 
Using fundamental theorem of algebra we can argue that LHS = RHS.


v.  Simplify a little

http://quicklatex.com/cache3/a6/ql_998e591ab64a9a7373ad88d0227efca6_l3.png

Let it be equation (b)


vi.

From equation (a) and (b), comparing coefficient of x^2.

http://quicklatex.com/cache3/2c/ql_1a1a5392e7a6cad8023511c8897daa2c_l3.png


From this problem, Basel series was named as a variable for real number x. 

http://quicklatex.com/cache3/34/ql_80f5ce2ee59772b8eb5be24f9f9a0834_l3.png

Also, later Riemann changed the domain of variable and named it Riemann zeta function. 

http://quicklatex.com/cache3/7e/ql_1d82d4edb3f6644d3f9bf18486488b7e_l3.png

This followed his paper in number theory and hypothesis which is famous called **Riemann Hypothesis**. It is still an unsolved problem. 

<hr>

Sources:

1. [Prime Obsession](https://www.goodreads.com/book/show/218392.Prime_Obsession)

2. [An infinite series surprises](https://plus.maths.org/content/infinite-series-surprises)

3. [Stem](https://studyinthestates.dhs.gov/2011/09/what-does-stem-stand-for)

4. [QuickLatex](http://quicklatex.com/)