ITPEC FE Subject B April 2024 Question 16

Maclaurin series approximation of sin(x) using iterative term computation.

The Maclaurin series for sin(x) is:

sin(x) = x/1! - x³/3! + x⁵/5! - ...

Each term relates to the previous one by a ratio. If vn is the current term and k tracks the factorial denominator (1, 3, 5, 7...), then after updating k ← k + 2, the next term is:

vn_new = -vn × x² / ((k - 1) × k)

Verification:
- When k goes from 1 → 3: ratio = -x² / (2 × 3) = -x² / 6 ✓ (converts x/1! to -x³/3!)
- When k goes from 3 → 5: ratio = -x² / (4 × 5) = -x² / 20 ✓ (converts -x³/3! to x⁵/5!)

The loop continues while abs(vn) > epsi — meaning we keep adding terms until the current term's absolute value drops below the threshold 1×10⁻⁷.

Blank A = > epsi (loop continues while term is still significant)
Blank B = -vn × x² ÷ ((k - 1) × k) (recurrence relation between consecutive terms)

Why not others:
- (a–e) with `≤ epsi` — This would exit immediately when the term is still large, producing an inaccurate result. The loop must continue while the term exceeds the threshold.
- (f) `-vn × x ÷ k` — Missing one factor of x and one factorial term; incorrect recurrence.
- (g) `-vn × x² ÷ k²` — Uses k² instead of (k-1)×k; gives wrong factorial progression.
- (i) `-vn × xᵏ ÷ k²` — Raises x to the power k, which doesn't match the term-to-term ratio.
- (j) `-vn × xᵏ ÷ ((k-1) × k)` — Same factorial part is correct but xᵏ is wrong; only x² is needed per step.

Key rule: In Maclaurin series, compute each term from the previous one using the ratio v(n+1)/v(n) rather than recalculating from scratch — track the changing denominator via an incrementing index.

AI-generated — may contain errors