Exact Value and Truncation Error Analysis for \( x = 0.8 \):

The exact value of the function is:

$$f(0.8) = (0.8)^3 = 0.512.$$

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 $$-2(6-\pi^2)\sin(0.8) \approx 5.55177$$ 5.03977
2 $$-2(6-\pi^2)\sin(0.8) + \frac{6-4\pi^2}{4}\sin(0.8) \approx -2.81427$$ 3.32627
3 $$1.3299$$ 0.817897


For more details, please contact me here.