In Prob. 26 Using e^x < e (0 < x < 1), conclude that |I_n| ≤ e/(n + 1) → 0 as n → ∞. Solve the iteration formula for I_n-1 = (e - I_n)/n, start from I₁₅ ≈ 0 and compute 4S values of I₁₄, I₁₃, · · ·, I₁.

Data from Prob. 26

Integrating by parts, show that I_n = ∫¹₀ e^xxⁿ dx = e -nI_n-1, I₀ = e - 1.

(a) Compute I_n, n = 0, · · ·, using 4S arithmetic, obtaining I₈ = -3.906. Why is this nonsense? Why is the error so large?

(b) Experiment in (a) with the number of digits k > 4. As you increase k, will the first negative value n = N occur earlier or later? Find an empirical formula for N = N(k)