Question

Prove that if \(|x|<1\) \(\frac{x^{3}}{1.3}-\frac{x^{5}}{3.5}+\frac{x^{7}}{5.7}-\ldots=\frac{1}{2}\left(1+x^{2}\right) \tan ^{-1} x-\frac{1}{2} x\)

Short answer

Question: Prove that \(\frac{1}{2}\left(1+x^{2}\right)\tan ^{-1} x-\frac{1}{2} x=\frac{x^{3}}{1.3}-\frac{x^{5}}{3.5}+\frac{x^{7}}{5.7}-\cdots\) Solution: The given series can be evaluated by connecting it to the power series representation of \(\tan^{-1}x\). When the power series of \(\tan^{-1}x\) is multiplied by \((1+x^2)\) and differentiated with respect to x, we obtain a series related to the given series. By comparing the resulting series with the given series, we can conclude that \(\frac{1}{2}\left(1+x^{2}\right)\tan ^{-1} x-\frac{1}{2} x=\frac{x^{3}}{1.3}-\frac{x^{5}}{3.5}+\frac{x^{7}}{5.7}-\cdots\).

Step-by-step solution

Recognize the given series as an alternating series

The given series is an alternating series because the terms alternate between positive and negative. Additionally, notice that the terms in the series involve a product of the form \(x^{2k+1}/(2k+1)(2k-1)\) for \(k = 1, 2, 3,\ldots\). This is important for when we construct the power series representation of the function.

Construct the power series representation of the function

We know that \(\tan^{-1}x\) can be represented as a power series \(\tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\) with radius of convergence 1. To reach our desired representation, we need to manipulate this power series.

Multiply the power series representation by \((1+x^2)\) and differentiate with respect to x to construct the series

First, we multiply the power series representation of \(\tan^{-1}x\) by \((1+x^2)\). This will give: \(\tan^{-1}x(1+x^2) = x + x^3 - \frac{x^5}{3} + \cdots\) Next, let's differentiate both sides with respect to x: \(\frac{d}{dx}((1+x^2)\tan^{-1}x) = \frac{d}{dx}(x + x^3 - \frac{x^5}{3} + \cdots)\) Applying the product rule for differentiation on the left-hand side, we get: \((1 + 2x)\tan^{-1}x + (1+x^2)\frac{1}{1+x^2} = 1 + 3x^2 - \frac{5x^4}{3} + \cdots\) Since the series for \(\tan^{-1}x\) converges and it is multiplied by \((1 + x^2)\) and differentiated with respect to x, the resulting power series converges point-wise to the series on the right-hand side.

Observe that the right-hand side is related to the given series

Now, observe that the right-hand side of the equation above is equal to the given series where the terms are multiplied by some even factors. We can rewrite the right-hand side as the sum of two series: \(1 + 3x^2 - \frac{5x^4}{3} + \cdots = \left(1 - \frac{x^5}{3} \frac{5}{3} + \cdots \right) + \left(3x^2 - \frac{5}{3}x^4 + \cdots \right)\)

Complete the proof by comparing the given series with the right-hand side

Now, compare the two series on the right-hand side with the given series: \(\frac{x^3}{1.3} - \frac{x^5}{3.5} + \frac{x^7}{5.7} - \cdots\) We can notice that multiplying the first series by \(\frac{1}{2}x\) and the second series by \(\frac{1}{2}(1+x^2)\) will give us the given series. Therefore, we can write: \(\frac{1}{2}x(1 - \frac{x^5}{3} \frac{5}{3} + \cdots)+\frac{1}{2}(1+x^2)(3x^2 - \frac{5}{3}x^4 + \cdots) = \frac{x^3}{1.3} - \frac{x^5}{3.5} + \frac{x^7}{5.7} - \cdots\) Finally, we can rewrite this equation in terms of the function as: \(\frac{1}{2}\left(1+x^{2}\right)\tan ^{-1} x-\frac{1}{2} x=\frac{x^{3}}{1.3}-\frac{x^{5}}{3.5}+\frac{x^{7}}{5.7}-\cdots\) This completes the proof.

Key concepts

Power Series Representation

A power series is an infinite sum of terms that are powers of a variable, often denoted as 'x', and multiplied by coefficients. It is represented as \(\sum_{n=0}^{\text{\infinity}} a_n x^n\). The coefficients \(a_n\) dictate the behavior and nature of the power series. Understanding the power series is crucial because many functions can be expressed as a power series, which enables calculations and predictions about their behavior.

For example, the arctangent function, \(\tan^{-1}x\), has a power series representation that starts \(x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots\). This representation is particularly valuable because it allows us to integrate and differentiate the function term by term, a trick that proves useful when dealing with more complex mathematical problems like the one in our exercise.

When constructing a power series representation, coherence with known series is often the key. In the provided exercise, adjusting the power series of \(\tan^{-1}x\) and leveraging the properties of power series allows us to express a seemingly complicated function in terms of a series we understand well. This leads to valuable insights into the nature of the function, especially within its interval of convergence.

Radius of Convergence

The radius of convergence of a power series is the value that defines the interval within which the series converges to a function. For a series centered around a point \(a\), the interval of convergence can be from \(a - R\) to \(a + R\), where \(R\) is the radius of convergence. If you plug any \(x\) within this interval into the power series, the series will converge; outside this interval, the series might diverge or its behavior becomes indeterminate.

In our exercise, we're working with a power series with a radius of convergence of 1, which means it converges for all \(x\) such that \(|x| < 1\). This detail is crucial, as the exercise assumes this condition to prove the given series. If this condition wasn't met, our series might not behave as expected, making our proof invalid. Understanding where a series converges is foundational to ensuring the validity of operations performed on the series, such as differentiation or integration.

Product Rule for Differentiation

The product rule is a fundamental technique in calculus used when differentiating the product of two functions. The rule states that for two differentiable functions \(u(x)\) and \(v(x)\), their derivative is given by \(u'(x)v(x) + u(x)v'(x)\). This rule allows us to systematically tackle differentiation problems that involve products of functions.

Applying the product rule can be particularly challenging when dealing with complex functions or series, as it requires careful manipulation and organization of terms. In our exercise, the product of \(1+x^{2}\) and \(\tan^{-1}x\) is differentiated, and the product rule is essential here. Each term in the power series has to be differentiated in turn, combined with the corresponding term from the other function, to create a new series that accurately represents the derivative. Without applying the product rule appropriately, we could not arrive at the correct function or series that the original exercise is trying to prove.

By mastering the product rule, students can solve differentiation problems involving series and functions that would otherwise be too complex to handle. It's crucial to practice this rule consistently to ensure accuracy when faced with similar challenges.