HMMT 二月 2001 · CALC 赛 · 第 7 题

HMMT February 2001 — CALC Round — Problem 7

Find the coefficient of x in the Maclaurin series (i.e. Taylor series around x = 0) for 1 . 2 1 − 3 x +2 x ∞ ∑ − 1 2

解析

Find the coefficient of x in the Maclaurin series (i.e. Taylor series around x = 0) for 1 . 2 1 − 3 x +2 x Solution: If you know formal power series, then this is not such a hard question, but 1 / 2 1 1 since this is a calculus test... Use partial fractions to get = − . Now each 2 1 − 3 x +2 x 1 − 2 x 1 − x of these can be expanded as a geometric series (or take derivatives and get the same result) 1 2 3 2 3 n n − 1 to get (1 + 2 x + 4 x + 8 x + · · · ) − (1 + x + x + x + · · · ), so the coefficient of x is 2 − 1. 2 When n = 12, that’s 2047 . ∞ ∑ − 1 2