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A power series is essentially an infinite sum of terms involving powers of a particular variable. In this context, a power series can be used to represent complicated functions in a simpler form. For example, consider the function \( an^{-1}(x)\). We can express it as a power series:

• \( an^{-1}(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} \)

This representation is valid for \(|x| \leq 1\), meaning the function can be accurately approximated by the series within this range. The power series provides an easier way to perform calculus operations, such as integration and differentiation.
The power series representation lets us explore different functions by substituting variables. For instance, in the case of \( an^{-1}(x^2)\), we substitute \(x^2\) for \(x\), leading to a new series representation. Using power series expansions helps us turn complex integrals into manageable algebraic sums.

The radius of convergence is a key concept that tells us the range of values for which a power series converges to a function. When working with series like that of \( an^{-1}(x)\), we need to determine where the series will accurately represent the function.To identify the radius of convergence for a power series, we typically look at the original series and apply certain convergence tests like the ratio test. In the case of \(\tan^{-1}(x^2)\), its power series shares the same convergence condition as its original form:

• \( |x| \leq 1 \)

This result tells us that the power series converges inside and on the boundary of the unit circle in the complex plane. Therefore, the radius of convergence is \(R = 1\). Such knowledge assures us that operations performed within this radius, like integration, will yield valid results.

Term-by-term integration is a technique that allows us to integrate each term of a power series individually. This can dramatically simplify the process of finding the integral of a complex function.When dealing with the series for \( an^{-1}(x^2)\), we can integrate the series one term at a time. If \(\sum_{n=0}^{\infty} a_n x^n \) is the power series representation of a function, its integral is obtained as:

• \( \sum_{n=0}^{\infty} \int a_n x^n \, dx = \sum_{n=0}^{\infty} \frac{a_n x^{n+1}}{n+1} + C \)

In practice, this means we simply integrate each coefficient and term independently, resulting in a neat series expression for the integral.For \(\int \tan^{-1}(x^2) \, dx\), each term of the power series for \(\tan^{-1}(x^2)\) is integrated separately, followed by summation. This leads to a new power series that represents the integral, allowing us to express the antiderivative accurately. The constant \(C\) is added at the end to represent the indefinite nature of the integral.