Sum of an Alternating Series Involving Pi

Find the sum $\sum_{n=0}^{\infty} \frac{(-1)^n \pi^{2n+1}}{4^{2n+1} (2n+1)!}$ and evaluate it.

This problem involves the calculation of an infinite series that takes the form of an alternating series. The series contains terms involving powers of pi and factorials in the denominator, hinting at potential connections to well-known mathematical constants or functions. One of the key concepts at play here is the representation of functions as series, particularly using Taylor and Maclaurin series. These series allow us to express transcendental functions, like sine or cosine, as infinite sums of polynomial terms. Recognizing the form of the terms in this series may evoke a familiar pattern, specifically the Taylor series expansion for trigonometric functions, which often include alternating signs and factorial terms in the denominator.

Understanding alternating series is crucial in evaluating this sum, especially with convergence properties. Alternating series can be evaluated using the Alternating Series Test, which ensures convergence if the terms decrease in absolute value and approach zero. Furthermore, recognizing the series' representation could link to known results, such as the arcsine or sine function. Evaluating such a series requires careful consideration of convergence issues and might involve manipulating the series or recognizing a pattern to simplify to known results. This exercise exemplifies applications of series in evaluating difficult summations, reinforcing the importance of pattern recognition and the utility of series in calculus.