Question

In Exercises 1 and 2 , state where the power series is centered. $$ \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} $$

Short answer

The power series is centered at \(\pi\).

Step-by-step solution

Identifying the Power Series

Firstly, identify that the provided expression is a power series. In this case, we have the power series given as: \\[ \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} \\]

Identifying the Center

The center of the power series is the value of \(x\) that makes the series equals zero. In the provided power series, \\(x-\pi\\) is zero when \(x = \pi\). Thereby, the center of the power series is \(\pi\).

Summary

Having worked through the steps, it can be concluded that the given power series is centered at \(\pi\).

Key concepts

Calculus and Power Series

Calculus is a branch of mathematics that deals with rates of change and accumulation. When it comes to power series, calculus provides tools to understand how functions can be represented as an infinite sum of their derivatives at a certain point, known as a series expansion. The power series \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} \) is such an expansion around the point \(x = \pi\).

Understanding power series is crucial in calculus as they offer a way to approximate complex functions with polynomials, which are easier to manipulate analytically. The coefficients of the terms in a power series can tell us about the behavior and properties of the original function. In this case, the power series represents a function that is centered around \( \pi \) - meaning that it can be used to approximate the function accurately in the neighborhood of \( \pi \) using the terms of the series.

Series Convergence

Series convergence is a concept in calculus that refers to the behavior of an infinite series. Specifically, it addresses whether the series approaches a finite value as the number of terms increases indefinitely. Convergence is an essential consideration when working with power series like \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} \).

There are several tests to determine convergence, including the Ratio Test, the Root Test, and the Comparison Test. For the given power series, which is in the form of an alternating series with factorial in the denominator, convergence can typically be expected because factorial growth helps to decrease the magnitude of the terms rapidly. To confirm convergence, however, one would usually apply one of these tests. Convergence is crucial for practical applications, as it ensures that using the power series as an approximation is meaningful and that the approximation will improve as more terms are added.

Factorial Notation

Factorial notation is instrumental in mathematics, particularly in combinatorics and series representations. The factorial of a non-negative integer \( n \) is the product of all positive integers less than or equal to \( n \), denoted by \( n! \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

In a power series, such as \( \sum_{n=0}^{\infty} \frac{(-1)^{n}(x-\pi)^{2 n}}{(2 n) !} \), factorial notation manifests in the denominator and serves to control the growth of the terms, facilitating convergence as \( n \) becomes large. The presence of the factorial term usually indicates a smooth, well-behaved function, as factorials grow very quickly and thus the terms in the series diminish in size. Understanding factorial notation is important when manipulating series and interpreting their convergence behavior.