A simple question

hypsin_0 · 2025-07-10 16:05:52

I know that 1/1^4+1/2^4+1/3^4+......is equal to π^4/90,but how to prove it?
I tried to solve it in my final exam in grade seven,but I failed——I got a wrong answer

KerimF · 2025-07-10 16:54:59

Are you sure that:
1/1^4+1/2^4+1/3^4+...... = n^4/90 ?
Let us check it for n=1, 2, 3, 4...

I think there is mistyping.

hypsin_0 · 2025-07-10 17:00:41

π^4/90,it's π,not n
I mean,add infinity terms (any grammar mistake?)

ktesla39 · 2025-07-10 17:53:39

This can be solved using calculus but to keep things easier, I just have a way: calculate everything from both LHS and RHS.
LHS = 1 + 1/16 + 1/81 + 1/256...... (notice that's decreasing rapidly when we increase n. So few numbers might give correct result. I'll use only 8 terms:)
= 1 + 0.0625 + 0.0123 + 0.00390625 + 0.0016 + 0.0007716049 + 0.000416493 + 0.0002441406
= 1.0815
RHS = 1.082

They are found to be equal up to 3 decimals and if you want more accuracy use more digits.

Was this helpful?

hypsin_0 · 2025-07-10 19:50:19

well...you can use calculus,maybe I can understand

ktesla39 · 2025-07-10 19:58:19

Not actually because I'm just few years ahead of you and still I don't know much about Calculus. The only thing I know is that this can be proved using Taylor Series which is really complex. ?

Bob · 2025-07-10 20:04:01

Wow! This is not a simple question. It needs a huge leap forward in advanced mathematics.

Ref: https://en.wikipedia.org/wiki/Riemann_zeta_function

I couldn't even get Wolfram Alpha to understand the question

Bob

I'll think about 'Taylor' to see if I can find an easy way to do this.

KerimF · 2025-07-10 20:11:42

For instance, I did the following on my old excel.

pi^4/90 is displayed as 1.082323233710860000000000000000

Sum for n=9740 or higher is displayed always as 1.082323233711140000000000000000

And, the excel error limit "(Sum - Constant)/Sum" becomes 0.000000000000255623799914129000

Last edited by KerimF (2025-07-10 20:12:31)

hypsin_0 · 2025-07-10 20:20:19

oops!I can't open this website.But...I don't want to use zeta function.

ktesla39 · 2025-07-10 20:24:53

I am in class 10 and I've started learning calculus (class 11) and after some time, when I'll find myself ready, I'll try learning Taylor Series and the Reimann zeta function. once I had a question something like

. I couldn't solve this and when I asked someone about it, they replied that this was a non elementary function an is written as Ei(x). That time I first saw Gamma and Zeta function.

ktesla39 · 2025-07-10 20:41:28

After searching some pages of my copy, the question was:

which became :

Maybe I had some mistakes.... but the result was non elementary function.

Bob · 2025-07-10 22:05:28

hi hypsin_0

Who set the question? And did they expect you to produce an answer straight away, or was research allowed?

Maybe they just wanted to show you that maths has lots of interesting branches.

Taylor hasn't given me an ideas yet. The Taylor series expansion has increasing powers (to infinity) and your problem just has 4th powers. Mmmm???

Bob

hypsin_0 · 2025-07-10 23:18:17

Euler do research on it.He also solved second power(not too hard),sixth power and eighth power.

ktesla39 · 2025-07-11 14:51:39

Ok now I don't seek any way to prove the equation. I tried dozens of methods and the only thing I can say is -- This cant be proved using pure algebra. I may be wrong but I'll be still good if we get any solution.

hypsin_0 · 2025-07-11 16:27:39

Maybe Euler's example can give you idea.
Basel problem
A mathematician try to solve

He added 1000 terms and got a number around 1.64,and he asked for help.Many mathematicians tried,but they all failed.
One day,Euler saw the approximation,he guess it's π^2/6.So he started doing research on sin(x)
sinx maybe=
......(so many things to write!)
and he got sinx/x is
we know 's quadratic coefficient is and sinx also equal to(you know,right?)
so we know -1/3!=
so is π^2/6

Last edited by hypsin_0 (2025-07-11 16:41:04)

ktesla39 · 2025-07-11 16:36:27

I tried 3 time with the Basel Problem:

I tried playing with it but still got nothing. I also tried learning Taylor, Fourier and Power series but all were complex.

This equation and our equation are related because they are terms of same pattern but you CANNOT find another using the first one. I only have a hope if Bob or any of us solves it.

hypsin_0 · 2025-07-11 16:41:32

so many things to write!It's so hard for me.

hypsin_0 · 2025-07-11 16:43:19

visit this website,it has the sollution of Basel problem.https://m.bilibili.com/video/BV11efiYaENR

Bob · 2025-07-12 00:30:35

Watch https://www.youtube.com/watch?v=GiWL7iKPoLM

Bob

ktesla39 · 2025-07-12 02:28:23

Thanks Bob!

I'll definitely love that.

Tomorrow I have exam of Optional Mathematics and I'll ask my teacher about the question. Maybe he can help.

Bob · 2025-07-12 03:21:18

Hope the exam goes well for you.

'Optional'. Hhhmmm! Interesting!

Bob

ktesla39 · 2025-07-12 03:36:47

Thanks!

We have two subjects Compulsary math and Optional maths.

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