Sum of 1/n^4 (Fourier Series & Parseval's Theorem)

TL;DR

Learn how to use Fourier series and the pacifier theorem to find the sum of a function.

Transcript

okay speedy I'll show you guys how to use the Fourier series and also the pacifier theorem to find the sum as n goes from 1 to infinity of 1 over and to the fourth power but because black piripi has a video on the Fourier series already Pichet gets go check that out I will have a link in the description for your convenience and now let me just put ... Read More

Key Insights

• 👨‍💼 The Fourier series can be used to break down a periodic function into a sum of sine and cosine functions.
• 🍹 The pacifier theorem establishes a connection between integrating f(x) squared and finding the sum of a function.
• ☺️ By choosing the function x squared, the video demonstrates the step-by-step process of finding the sum using Fourier series and the pacifier theorem.
• ✖️ The coefficients a0, an, and bn are calculated by integrating the function multiplied by cosine(nx) and sine(nx) over one period.
• 🍹 The pacifier theorem equation, derived from integrating f(x) squared, helps in finding the sum of a function.
• 🍹 The pacifier theorem simplifies the calculations by relating the coefficients to the sum directly.
• ☺️ By plugging in the function x squared into the calculations, the video demonstrates how to find the sum using Fourier series and the pacifier theorem.

Questions & Answers

Q: What is the Fourier series and how is it related to finding the sum of a function?

The Fourier series decomposes a periodic function into a sum of sine and cosine functions. By using the pacifier theorem, we can relate the coefficients of the Fourier series to the sum of a function.

Q: How do you find the coefficients a0, an, and bn in the Fourier series?

To find a0, we integrate the function f(x) over one period and divide by the period length. For an and bn, we integrate f(x) multiplied by cosine(nx) and sine(nx) respectively over one period and divide by the period length.

Q: How is the pacifier theorem used to find the sum of a function?

The pacifier theorem states that integrating f(x) squared over one period multiplied by pi equals 2*a0^2 plus the sum of (an^2 + bn^2). By calculating the coefficients a0, an, and bn, we can find the sum of the function.

Q: What function is used in the video to demonstrate finding the sum using Fourier series?

The function f(x) used in the video is x squared. By plugging this function into the calculations, the video shows how to find the sum.

Summary & Key Takeaways

• The video explains how to use the Fourier series and the pacifier theorem to find the sum of a function.

• The pacifier theorem establishes a relationship between integrating f(x) squared and finding the sum of a function.

• By choosing the function f(x) as x squared, the video shows step-by-step calculations to find the sum.