Sum of an Alternating Series Involving Pi2

Find the sum $\sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n+1}}{3^{2n-2} (2n)!}$.

This problem delves into the fascinating world of infinite series, where we will explore the sum of the given alternating series involving pi. To tackle this problem, one needs to understand the concept of alternating series and how they converge. Alternating series are ones where consecutive terms have opposite signs, typically involving a negative one raised to the power of the term index. Recognizing this form is crucial in identifying the nature of convergence of a series.

The series given here includes a factorial term and powers of pi. Such series often resemble known series, and recognizing this form might reveal familiar patterns. In particular, the structure involving factorials hints at a relationship with Taylor or Maclaurin series, commonly used to represent functions as infinite sums. Knowing the basics of Taylor series expansion is invaluable in this scenario as it allows one to match the series given to a known function or its modifications, ultimately aiding in finding the sum of the series.

Furthermore, the convergence criteria play a pivotal role. For alternating series, the Alternating Series Test is a handy tool. It states that if the absolute value of the terms decreases monotonically and approaches zero, the series converges. This problem is an excellent exercise in applying these concepts to determine the series' sum, leveraging both recognition of series forms and convergence tests to successfully navigate to a solution.