
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Decompose into a Fourier series the periodic functions:
1) \( f(x)=x^{2} \) for \( -\pi \leq x < \pi \) with \( f(x)=f(x+2\pi) \).

Using the resulting decomposition, find the sum of the series:
a) \( 1-\frac{1}{2^{2}}+\frac{1}{3^{2}}-\frac{1}{4^{2}}+\ldots \)
b) \( 1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\ldots \)

2) \( \varphi(x)=x^{2} \) for \( 0 < x < \pi \) with \( \varphi(x)=\varphi(x+\pi) \).

3) \( u=\cos \frac{x}{2} \) for \( 0 < x \leq 2\pi \) with \( u(x)=u(x+2\pi) \).

Present the answer in LaTex format: \boxed{Your answer}
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