Math Problem Statement
Solution
To solve the problem, we need to find the Fourier series representation of the function on the interval . From this, we will deduce that:
Step 1: General form of Fourier Series
The Fourier series of a function defined on the interval is given by:
where the Fourier coefficients and are calculated as follows:
Step 2: Calculating the coefficients for
1. (Constant term):
So,
2. (Cosine terms):
We will use integration by parts here. This integral evaluates to:
3. (Sine terms):
Since is an odd function over , the integral evaluates to zero, so:
Step 3: Fourier series representation
Now, the Fourier series representation of is:
Step 4: Deducing the series
To deduce the series for , set in the Fourier series expansion:
Since , the series becomes:
Rearranging gives:
Now subtract to form the alternating series given in the question:
This is the desired result.
Would you like more details on any step? Here are five related questions:
- How does integration by parts work for calculating Fourier coefficients?
- What is the significance of and terms in Fourier series?
- Can the Fourier series be used to approximate discontinuous functions?
- How would the Fourier series change if was periodic with a different period?
- What other mathematical results can be derived from the Fourier series of basic functions?
Tip: Always check for symmetry in functions when calculating Fourier coefficients, as this can simplify calculations (e.g., odd or even functions leading to zero sine or cosine terms).
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Math Problem Analysis
Mathematical Concepts
Fourier Series
Trigonometric Functions
Series Representation
Formulas
Fourier series general form: f(x) = a_0/2 + Σ (a_n cos(nx) + b_n sin(nx))
a_n = (1/π) ∫(0 to 2π) f(x) cos(nx) dx
b_n = (1/π) ∫(0 to 2π) f(x) sin(nx) dx
π^2 / 12 = 1 / 12 - 1 / 2^2 + 1 / 3^2 - ...
Theorems
Parseval's Theorem
Orthogonality of Trigonometric Functions
Suitable Grade Level
College-level or Advanced High School
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