A-Level Maclaurin series: worked solution

Question

Find the first three non-zero terms of the Maclaurin expansion of e^x sin(x + π).

Method: do not expand sin(x + π) as a series from scratch. Simplify the trigonometric factor first with the addition formula, then multiply the two standard Maclaurin series from the formula list. Step 1 - simplify sin(x + π). sin(A + B) = sin A cos B + cos A sin B, with A = x and B = π: sin(x + π) = sin x cos π + cos x sin π cos π = −1 and sin π = 0, so sin(x + π) = −sin x Therefore e^x sin(x + π) = −e^x sin x. Step 2 - write down the two standard series. e^x = 1 + x + x²/2! + x³/3! + ... = 1 + x + x²/2 + x³/6 + ... sin x = x − x³/3! + ... = x − x³/6 + ... Step 3 - decide how far to go. The product starts at x (the sin series has no constant term), so the first three non-zero terms will be the x, x² and x³ terms. Keep everything up to x³ and discard the rest. Step 4 - multiply out, collecting by power of x. (1 + x + x²/2 + x³/6 + ...)(x − x³/6 + ...) x terms: 1 × x = x x² terms: x × x = x² x³ terms: (x²/2) × x = x³/2, and 1 × (−x³/6) = −x³/6 total x³/2 − x³/6 = 3x³/6 − x³/6 = 2x³/6 = x³/3 So e^x sin x = x + x² + x³/3 + ... Step 5 - apply the minus sign from Step 1. e^x sin(x + π) = −(x + x² + x³/3 + ...) = −x − x² − x³/3 + ... The first three non-zero terms are −x, −x² and −x³/3.