sin5x ..

if f(x)= x 2 (0 x 2p)
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To find the Fourier series of the function f (x) = x2 on the interval 0 ≤ x ≤ 2π with period 2π , we
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Step 1: Fourier Series Formula

Fourier series approximation The Fourier series of a periodic function f (x) is given by:

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f (x) = a0 + ∑ (an cos(nx) + bn sin(nx)) ,
n=1
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where the coefficients are calculated as:
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2π
1
Statistics & Scales Explained a0 = ∫ f (x) dx,
2π 0
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1 2π
Chebyshev Theorem Explained an = ∫ f (x) cos(nx) dx,
π 0
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1 2π
bn = ∫ f (x) sin(nx) dx.
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Step 2: Calculate a0
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2π
1
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a0 = ∫ x2 dx.
2π 0
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The integral is:
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x3
Photo Assistance Needed ∫ x2 dx = .
3
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Evaluate over the limits:
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1 (2π)3 03 1 8π 3 4π 2
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Step 3: Calculate an
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1 2π 2
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an = ∫ x cos(nx) dx.
π 0
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Using integration by parts:
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sin(nx) sin(nx)
xsinxcosnx identity ∫ x2 cos(nx) dx = x2 − ∫ 2x dx.
n n
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Continue integrating by parts for ∫ x sin(nx) dx. The result will involve terms proportional to
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cos(nx) and sin(nx). After detailed calculations (which are tedious but follow standard integration
techniques), an evaluates to:

−4 −4
an = cos(2nπ ) = 2 .
n2 n

Step 4: Calculate bn

1 2π 2
bn = ∫ x sin(nx) dx.
π 0

Similarly, using integration by parts, it can be shown that:

bn = 0.
2
This is because x sin(nx) is an odd function over the interval 0 to 2π .

Step 5: Fourier Series

The Fourier series for f (x) = x2 is:
∞
4π 2 4
f (x) = − ∑ 2 cos(nx).
3 n=1
n

Step 6: Partial Sums up to cos(5x)

The partial sum up to cos(5x) is:

4π 2 cos(x) cos(2x) cos(3x) cos(4x) cos(5x)

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