Todd and Rodd

Calculus Level 5

$\begin{aligned} f_n(x) &=& \tan(f_{n-1}(x)) \quad, \quad f_1 = \tan(x) \\ g_n(x) &=& \arctan(g_{n-1}(x)) \quad,\quad g_1 = \arctan(x) \end{aligned}$

Given the two recurrence relations above for $n=2,3,4,\ldots$ . Evaluate

$\large \lim_{x\to0} \dfrac{f_{2016}(x) - g_{2016}(x) }{x^3}.$

The answer is 1344.

1 solution

Otto Bretscher
Dec 19, 2015

It is not hard to see, by induction on $n$ , that the Taylor series of $f_n(x)$ is $x+\frac{n}{3}x^3+...$ while that of $g_n(x)$ is $x-\frac{n}{3}x^3+...$ . Thus the limit we seek is $\frac{2\times n}{3}=\boxed{1344}$ for $n=2016$ .

Todd and Rodd

The answer is 1344.

1 solution

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