Chapter 1: Q18MP (page 1)

Find the Maclaurin series of the following functions.
$arc t a n x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$

Short Answer

Expert verified

Maclaurin series of $\arctan x is x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots . .$

Step by step solution

The Maclaurin series and the given function.

Function is $\arctan x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$

The Maclaurin series of $\int_{0}^{x} \frac{d u}{1 + u^{2}}$ is given as follows:

$\int_{0}^{x} \frac{d u}{1 + u^{2}} = \int_{0}^{x} (1 - u^{2} + u^{4} - u^{6} + \dots . .) d u \int_{0}^{x} \frac{d u}{1 + u^{2}} = {(u - \frac{u^{3}}{3} + \frac{u^{5}}{5} - \frac{u^{7}}{7} + \dots . .)}_{0}^{x} \int_{0}^{x} \frac{d u}{1 + u^{2}} = x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots$

Now, we have:

$\frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + \dots . .$

Calculation by the Maclaurin series.

Substitute $u^{2}$ for $x$ in expansion of $\frac{1}{1 + x}$ as follows:

$\begin{array}{rcl} \frac{1}{1 + u^{2}} & = & 1 - u^{2} + {(u^{2})}^{2} - {(u^{2})}^{3} + \dots . . \\ \frac{1}{1 + u^{2}} & = & 1 - u^{2} + u^{4} - u^{6} + \dots . . \end{array}$

Apply integration for $\lim$ $0$ to $x$ in the above equation as follows:

Thus, the Maclaurin series of $\arctan x is x - \frac{x^{3}}{3} + \frac{x^{5}}{5} - \frac{x^{7}}{7} + \dots . .$

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Find the Maclaurin series of the following functions.
$arc t a n x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$

Short Answer

Step by step solution

The Maclaurin series and the given function.

Calculation by the Maclaurin series.

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Find the Maclaurin series of the following functions.arctanx=∫0xdu1+u2

Short Answer

Step by step solution

The Maclaurin series and the given function.

Calculation by the Maclaurin series.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Electricity and Magnetism

Magnetism and Electromagnetic Induction

Waves Physics

Thermodynamics

Fluids

Radiation

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series of the following functions.
$arc t a n x = \int_{0}^{x} \frac{d u}{1 + u^{2}}$