Trigonometric way to find the solution of Basel problem

Let's generalise the whole thing first

Generally \[\sum_{n=1}^∞ \dfrac{\sin(n\theta)}{n} = \dfrac{π-\theta}{2} \] Where \(0<\theta≤π\)

And consequently n=1cos(nθ)n2=n=11n2πθ2+θ24=ζ(2)πθ2+θ24\sum_{n=1}^∞ \dfrac{\cos(n\theta)}{n^2} = \sum_{n=1}^∞ \dfrac{1}{n^2} -\dfrac{π\theta}{2} +\dfrac{\theta^2}{4} =\zeta(2)-\dfrac{π\theta}{2} +\dfrac{\theta^2}{4} refer this solution

let's take this sum n=1sin2(nθ)n2\displaystyle\sum_{n=1}^∞ \dfrac{\sin^2(n\theta)}{n^2} It is =n=1(12n2cos(2nθ)2n2)=\sum_{n=1}^∞\left( \dfrac{1}{2n^2} - \dfrac{\cos(2n\theta)}{2n^2} \right) =ζ(2)2(ζ(2)2+(2θ)28πθ2)= \dfrac{\zeta(2)}{2} -\left (\dfrac{\zeta(2)}{2} +\dfrac{(2\theta)^2}{8} -\dfrac{π\theta}{2}\right ) =θ(πθ)2= \dfrac{\theta(π-\theta)} {2} Now n=1sin2(nπ2)n2=π2(ππ/2)2=π28\sum_{n=1}^∞ \dfrac{\sin^2(n\frac{π}{2})}{n^2} = \dfrac{ \frac{π}{2} (π-π/2)}{2} = \dfrac{π^2}{8}     112+132+152+=π28\implies \dfrac{1}{1^2} +\dfrac{1}{3^2} +\dfrac{1}{5^2} +\cdots= \dfrac{π^2}{8} But this is all odd terms . ζ(2)4+n=01(2n+1)2=ζ(2)\dfrac{\zeta(2)}{4} +\sum_{n=0}^∞\dfrac{1}{(2n+1)^2} = \zeta(2) ζ(2)=43×π28=π26\zeta(2)= \dfrac{4}{3} × \dfrac{π^2}{8} = \dfrac{π^2}{6}

Note by Dwaipayan Shikari
1 week, 6 days ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...