Define the function for all real by

MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Define the function

h

for all real

x

h(x)=\dfrac{\arctan x}{\sqrt{1+x^{2}}}

. Assume

h

possesses derivatives of all orders at

x=0

. (a) Show that the Maclaurin series of

\arctan x

contains only odd powers of

x

, and write the first four nonzero terms of this series. (b) Show that the Maclaurin series of

\dfrac{1}{\sqrt{1+x^{2}}}

contains only even powers of

x

, and write the first four nonzero terms of this series. (c) Using the results from parts (a) and (b), derive an explicit formula for the coefficient of

x^{2n}

in the Maclaurin series of

h

. Express your answer in terms of

n

and factorials. (d) Hence obtain a closed‐form expression for

h^{(2n)}(0)

for every non-negative integer

n

. Justify each step of your derivation.

Related theory

Higher-Order Derivatives
Advanced Differentiation
Power Series
Maclaurin Series

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