The Basel Problem: A Solid Solution Using Derivatives and Integrals

[hover]

http://www.maa.org/news/howeulerdidit.html"
"[URL
[/URL]
I found a nice solution to the Basel problem on the internet which I am liking very much. It deals with derivatives and integrals and appears to be a much more solid solution to the Basel problem than Euler's original solution. There is just one thing I am having trouble with to understand the solution. I think it is just me being an idiot but I can't understand the end of the second page,

The series on the right is the sum of the reciprocal of the odd squares, tantalizingly close to the Basel problem, and an easy trick makes it into a solution. Any number is the product of an odd number and a power of 2. For odd numbers, the power of 2 is 2^0 . Hence, any square is the product of an odd square and a power of 4. So, Euler multiplies this equation by the sum of the reciprocals of the powers of 4, as follows:

It sounds like garble and I don't get the logic. Can someone explain to me?

Thanks!

 

[LCKurtz]

http://www.maa.org/news/howeulerdidit.html"
"[URL
[/URL]
I found a nice solution to the Basel problem on the internet which I am liking very much. It deals with derivatives and integrals and appears to be a much more solid solution to the Basel problem than Euler's original solution. There is just one thing I am having trouble with to understand the solution. I think it is just me being an idiot but I can't understand the end of the second page,

It sounds like garble and I don't get the logic. Can someone explain to me?

Thanks!

Are you referring to this quote?:

"Any number is the product of an odd number and a power of 2. For odd numbers, the power of 2 is 2^0 . Hence, any square is the product of an odd square and a power of 4. "

All it is saying is that if n is odd, n = 2⁰*n and n² = 4⁰*n², which isn't saying much.

If n is even, factor out all factors of 2 so n has the form n = 2^pk for some odd integer k. Then n²=2^2pk² = 4^pk².

 

[hover]

Are you referring to this quote?:

"Any number is the product of an odd number and a power of 2. For odd numbers, the power of 2 is 2^0 . Hence, any square is the product of an odd square and a power of 4. "

All it is saying is that if n is odd, n = 2⁰*n and n² = 4⁰*n², which isn't saying much.

If n is even, factor out all factors of 2 so n has the form n = 2^pk for some odd integer k. Then n²=2^2pk² = 4^pk².

I'm sorry but I'm still having a hard time understanding. I don't see what the small quote(what I quoted in my first post) is talking about or referring to. The equation above where I quoted says pi^2/8 = ∑ 1/(1+2 k)^2 (n = 0 to infinity). I don't see how euler knows to multiply by 4/3 to get pi^2/6 and solve the Basel problem. :(

 

[friz82]

I would try to answer to your question.
I had the same problems at that point when I read the text for the first time.
I think it can be explained as follows:
Every integer number, let's say 1,2,3,4... can be written as an odd number multiplied for some power of 2. If we take any odd number, let's say 7, we can have 7=7*2^0, 14=7*2^1 and so on.
The point is that if we start from the ensemble of just the odd numbers (1,3,5,7...) we can obtain the whole list of integer numbers by multiplying every term by the ensemble of powers of 2 (1,2,4,8...)
Therefore, to pass from the sum of the reciprocals of the odd squares to the sum of the reciprocal of all the integers we multiply the first sum by the sum of the reciprocal of the powers of 4 (2 squared). This last sum is equal to 3/4.
I hope this will be of some help

 

[CellCoree]

I am always intrigued by different approaches to solving problems. The Basel problem has been a topic of interest for centuries and it is exciting to see new solutions being proposed. The use of derivatives and integrals in this solution is a clever approach and shows the power of mathematical tools in solving complex problems.

To address your confusion about the end of the second page, let's break down the logic step by step. The goal is to find a way to manipulate the series on the right, which is the sum of the reciprocals of the odd squares, so that it becomes the solution to the Basel problem.

1. Any number can be written as the product of an odd number and a power of 2. For example, 10 can be written as 5 x 2^1.

2. For odd numbers, the power of 2 is always 2^0, which is just 1. So, for odd numbers, we can write the product as the number itself. For example, 5 can be written as 5 x 1.

3. Similarly, any square number can be written as the product of an odd square and a power of 4. For example, 25 can be written as 5^2 x 4^0.

4. Now, let's apply this logic to the series on the right. We can write each term in the series as the reciprocal of an odd square multiplied by a power of 4. For example, 1/9 can be written as 1/(3^2 x 4^0).

5. If we multiply the entire series by the sum of the reciprocals of the powers of 4, we get the following:

(1/1 + 1/4 + 1/16 + 1/64 + ...) x (1 + 1/4 + 1/16 + 1/64 + ...)

Note that the second term in the above equation is the sum of the reciprocals of the powers of 4.

6. Simplifying the above equation, we get:

(1/1 + 1/4 + 1/16 + 1/64 + ...) x (1 + 1/4 + 1/16 + 1/64 + ...) = 1 + (1/4 + 1/16 + 1/64 + ...) + (1/4 + 1/16