Let be analytic at with Maclaurin series

MATHEMATICS · Single correct · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Let ff be analytic at x=0x=0 with Maclaurin series f(x)=∑k=0∞akxkf(x)=\sum_{k=0}^{\infty}a_k x^{k}. Suppose (1+x+x22+x36+⋯ ) f(x)=1+x2+23x3+⋯\bigl(1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\cdots\bigr)\,f(x)=1+x^{2}+\frac{2}{3}x^{3}+\cdots for all xx near 00. What is f′′′(0)f'''(0)? (a) −6-6 (b) −3-3 (c) −12-\dfrac12 (d) 33

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