Exact Value and Truncation Error Analysis:

The exact value for \( x = \pi/3 \) is:

$$f(\pi/3) = |\pi/3| - \pi/2 = \pi/3 - \pi/2 = -\pi/6 \approx -0.52359878.$$

Truncation error for a partial sum \( S(x) \) is:

$$E(x) = |S(x) - f(x)|.$$

Table of Partial Sums and Errors:

Number of non-zero terms Value of the partial sum Truncation error
1 $$-\frac{4}{\pi}\cos(\pi/3) \approx -0.63661977$$ 0.11302100
2 $$-\frac{4}{\pi}\cos(\pi/3) - \frac{4}{9\pi}\cos(3\pi/3) \approx -0.49514871$$ 0.02845006
3 $$-0.52061350$$ 0.00298527


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