Problem 21 - Calculus

Question

Find an expression for the general term of the series and give the range of values for the index \((n \text { or } k\) for example). $$\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots$$

Step-by-Step Solution

Answer

The general term is \((-1)^n \frac{x^{2n+1}}{2n+1}\) for \(n = 0, 1, 2, \ldots\).

Step 1: Identifying the Pattern

The given series is \(\arctan x=x-\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}+\cdots\). Each term in this series can be written in the form \((-1)^n \frac{x^{2n+1}}{2n+1}\).

Step 2: General Term Formula

Based on the observed pattern, the general term of the series is represented as \((-1)^n \frac{x^{2n+1}}{2n+1}\) where \(n\) starts from 0. So, the general term is given by \(a_n = (-1)^n \frac{x^{2n+1}}{2n+1}\).

Step 3: Range of Index

The index \(n\) in the series starts from 0 and increases by 1 in each succeeding term. Therefore, \(n\) takes on non-negative integer values: \(n = 0, 1, 2, 3, \ldots\).

Key Concepts

Maclaurin Series

Maclaurin series is a specific type of Taylor series. It's named after Colin Maclaurin, a Scottish mathematician. A Maclaurin series is essentially a Taylor series expansion of a function about zero. This means it's a way to approximate functions using polynomials centered at zero.
The general form of a Maclaurin series for a function \( f(x) \) is:

• \( f(x) = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \frac{f'''(0)}{3!}x^3 + \ldots \)

In our exercise, we're dealing with the arctangent function, \( \arctan x \). Its series expansion, when derived from a Maclaurin series, is:

• \( \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \ldots \)

This is because the derivatives of \( \arctan x \) at zero follow a specific pattern that allows us to write it in this polynoimal form.

General Term

Understanding the general term of a series helps us find any term within the series. For the series given by \( \arctan x \), the observed pattern is key. We see it alternates in sign and increases in odd powers of \( x \), divided by the corresponding odd numbers.
The general term can be expressed as:

• \( a_n = (-1)^n \frac{x^{2n+1}}{2n+1} \)

Here, the \((-1)^n\) accounts for the alternating signs. The power of \( x \) within each term is expressed as \( 2n + 1 \), which ensures only odd powers appear. The denominator \( 2n + 1 \) matches the corresponding power of \( x \), ensuring that as \( n \) increases, the series follows its observed pattern.

Power Series

Power series often emerge in mathematics as useful tools for representing functions as sums of infinitely many terms. These series are expansions where each term is a power of the variable multiplied by a coefficient.
Consider a general power series expressed as:

• \( a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots \)

For \( \arctan x \), the power series is a sum of terms of the form \( \frac{x^{2n+1}}{2n+1} \), featuring coefficients that depend on simpler patterns.
The convergence of a power series, which means for which values of \( x \) the series accurately represents the function, is another crucial concept. For \( \arctan x \), this power series converges for \( -1 < x < 1 \). Understanding this range is important because it defines where the series can be applied to approximate \( \arctan x \) effectively.