Expressing Fraction as Power Series

Express $\frac{1}{1 + x^2}$ as the sum of a power series.

The problem of expressing the given fraction as the sum of a power series involves understanding the concept of power series expansion. Power series are infinite series of the form sum of a_n(x-c)^n, where a_n represents the coefficient of the nth term, c is the center of the series, and x is the variable. When representing functions as power series, particularly around a point, it allows us to express complex functions in terms of polynomials, which are easier to work with for both theoretical and practical applications.

In this specific problem, you'll want to recall the geometric series summation formula, which is a foundational concept when dealing with power series. The geometric series $\frac{1}{1-r}$ equals the sum of $r^n$, and converges when the absolute value of $r$ is less than 1. By understanding how to manipulate the function into a form that matches a geometric series or its derivatives, you can find a corresponding power series representation.

Additionally, consider the domain of convergence for the series, which in this kind of problem is often determined by examining where the expression inside the series lies within the -1 to 1 range. This problem not only tests comprehension of series and their manipulations but also reinforces skills in algebraic manipulation, a critical skill in mathematical problem-solving.