Express Function as Sum of Power Series Using Partial Fractions

Express the function as a sum of a power series by first using partial fractions, then determine the interval of convergence.

Expressing a function as a sum of a power series involves breaking down a complex rational function into simpler components, which can then be individually expanded into power series. The process of using partial fractions enables this breakdown by decomposing the rational function into a sum of fractions with simpler denominators. This process is particularly useful because each simpler fraction can be more readily transformed into a power series representation, allowing for the overall function to be expressed as a series.

Once the function is expressed in the form of partial fractions, each term can then be individually expressed as a power series. This involves expressing functions in terms of their power expansion using techniques such as polynomial long division or geometric series expansions, depending on the form of the partial fractions. It's crucial to consider the radius and interval of convergence for each term in the series since it defines the interval over which the power series representation is valid.

Determining the interval of convergence often involves techniques such as the ratio test or the root test, which help determine where the series converges absolutely. Understanding this interval is vital as it dictates where the power series representation of the function holds true, a key concept when working with series expansions in calculus.