Sum of 1/n^4 (Fourier Series & Parseval's Theorem) | Summary and Q&A

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May 26, 2019
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blackpenredpen
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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.

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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.

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