Problem 1 - Essential Calculus

Question

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

Step-by-Step Solution

Answer

The power series is centered at \(x=0\).

Step 1: Understand Power series

Recognize that a power series takes the form: \(\sum_{n=0}^{\infty} a_n (x-c)^n\). In our series \(\sum_{n=0}^{\infty} n x^{n}\), the \(a_n\) term is \(n\), and all \(x\) terms are in the form \(x^n\), which can be rewritten as \((x-0)^n\). This indicates that the series matches the general form of a power series.

Step 2: Identify the center

Based on comparison with the general form of a power series, where \(c\) is the center of the series, in \(\sum_{n=0}^{\infty} n (x-c)^{n}\), it can be seen that the \(x\) term in our power series is raised to the \(n\) power as \(x^n\), or \((x-0)^n\). Hence, \(c=0\) is the center of the power series.

Key Concepts

Series center

In the world of power series, the series center is a fundamental concept that helps us understand where the series is initially focused. Essentially, the center of a power series is a specific point around which the series is expanded. It is denoted as \(c\) in the general power series form \(\sum_{n=0}^{\infty} a_n (x-c)^n\).
The series center plays a crucial role in the behavior of the series, determining how it behaves near the point \(c\).
Identifying the center is straightforward once you recognize the general form of a power series:

• If the series is written as \(\sum_{n=0}^{\infty} a_n (x-c)^n\), the center is at \(c\).
• If it appears as \(\sum_{n=0}^{\infty} a_n x^n\), it implies \((x-0)^n\), indicating that the center is at 0.

For example, in the series \(\sum_{n=0}^{\infty} n x^n\), the term \(x^n\) can be rewritten as \((x-0)^n\), showing us that the center of the series is at 0. Understanding the center is vital for analyzing the convergence and behavior of power series.

General form of power series

The general form of a power series provides a clear framework for understanding and manipulating these mathematical expressions. This form is represented as \(\sum_{n=0}^{\infty} a_n (x-c)^n\), where:

• \(n\) is the index of summation and typically starts at 0.
• \(a_n\) are the coefficients that can vary depending on the term.
• \(x\) represents the variable.
• \(c\) is the center of the series, indicating where the series is centered on the x-axis.

This form allows us to express infinitely many terms succinctly and is crucial for evaluating functions that can be expanded around a point \(c\).
In many cases, power series are used to approximate complex functions by infinite sums of simpler terms. This characteristic is useful in calculus, where power series can represent functions within a certain radius of convergence. For example, a series such as \(\sum_{n=0}^{\infty} n (x-c)^n\) shows the general form with \(a_n = n\) and may be centered at any point \(c\), which in this particular example would be 0.

Convergence of series

The convergence of a series is a critical concept because it determines whether the infinite sum of the series results in a finite value. When dealing with power series, convergence informs us about the interval or domain where the series accurately represents a function.
Power series converge differently based on the value of \(x\) and its distance from the center \(c\). Understanding convergence involves a few key ideas:

• The radius of convergence \(R\), which defines the interval \((c-R, c+R)\) around \(c\), where the series converges.
• If \(x\) is within the radius of convergence, the series converges to a function value; outside this range, it diverges.
• The series may converge at the endpoints of this interval; however, this must be verified separately.

Finding the interval of convergence typically involves techniques such as the ratio test or root test. In series like \(\sum_{n=0}^{\infty} n x^n\), the convergence will depend on similar tests to ensure the series sums up to a fixed value for certain \(x\). Knowing where a series converges provides essential insights into its practical applications and limitations.