HMMT 二月 2001 · CALC 赛 · 第 7 题

英文解析

7. Find the coefficient of x in the Maclaurin series (i.e. Taylor series around x = 0) for
   .1
   1 − 3 x +2 x 2
   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
   1 − 3 x +2 x 1 − 2 x 1 − x 2
   of these can be expanded as a geometric series (or take derivatives and get the same result)
   2 3 2 3 n n − 11
   to get (1 + 2 x + 4 x + 8 x + · · · ) − (1 + x + x + x + · · · ), so the coefficient of x is 2 − 1.
   When n = 12, that’s 2047 .2
   ∞
   ∑
   − 1 2