The Fourier series representation of the following function { X f(x + 4) = f(x) is f(x)≈1 - Σm=1 π² f(x) = -X cos((2m-1) Tx/2) (2m-1)² - 2 < x < 0, 0 ≤ x < 2;
The Fourier series representation of the following function { X f(x + 4) = f(x) is f(x)≈1 - Σm=1 π² f(x) = -X cos((2m-1) Tx/2) (2m-1)² - 2 < x < 0, 0 ≤ x < 2;
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer all part correctly
![### Fourier Series Representation
The Fourier series representation of the function:
\[
f(x) =
\begin{cases}
-x, & \text{for } -2 < x < 0, \\
x, & \text{for } 0 \leq x < 2;
\end{cases}
\]
is given by:
\[
f(x+4) = f(x)
\]
\[
f(x) \approx 1 - \frac{8}{\pi^2} \sum_{m=1}^{\infty} \frac{\cos((2m-1)\pi x/2)}{(2m-1)^2}.
\]
#### Exercise
a) **Task:** State why (by using Fourier theorem) the Fourier series representation given above is a valid representation of \( f(x) \) on \(-2 < x < 2\).
b) **Task:** Show from the Fourier series representation given in part (a) that
\[
\frac{\pi^2}{8} = 1 + \frac{1}{3^2} + \frac{1}{5^2} + \ldots = \sum_{m=1}^{\infty} \frac{1}{(2m-1)^2}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2Fd51e3be9-fbe9-483c-93cd-182034665a93%2Fnkh4jki_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Fourier Series Representation
The Fourier series representation of the function:
\[
f(x) =
\begin{cases}
-x, & \text{for } -2 < x < 0, \\
x, & \text{for } 0 \leq x < 2;
\end{cases}
\]
is given by:
\[
f(x+4) = f(x)
\]
\[
f(x) \approx 1 - \frac{8}{\pi^2} \sum_{m=1}^{\infty} \frac{\cos((2m-1)\pi x/2)}{(2m-1)^2}.
\]
#### Exercise
a) **Task:** State why (by using Fourier theorem) the Fourier series representation given above is a valid representation of \( f(x) \) on \(-2 < x < 2\).
b) **Task:** Show from the Fourier series representation given in part (a) that
\[
\frac{\pi^2}{8} = 1 + \frac{1}{3^2} + \frac{1}{5^2} + \ldots = \sum_{m=1}^{\infty} \frac{1}{(2m-1)^2}.
\]
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