Problem 21 - Essential Calculus

Question

Use the power series $$\frac{1}{1+x}=\sum_{n=0}^{\infty}(-1)^{n} x^{n}$$ to determine a power series, centered at 0 , for the function. Identify the interval of convergence. $$ g(x)=\frac{1}{x^{2}+1} $$

Step-by-Step Solution

Answer

The function \(g(x)=\frac{1}{x^{2}+1}\) can be represented by the power series \(\sum_{n=0}^{\infty} x^{2n}\), and the interval of convergence is \(-1 \le x \le 1\).

Step 1: Replace \(x\) in the known series

Start by inserting \(-x^{2}\) for \(x\) in the known series for \(\frac{1}{1+x}\): \[g(x) = \sum_{n=0}^{\infty}(-1)^{n} (-x^{2})^{n} = \sum_{n=0}^{\infty}(-1)^n (-1)^n x^{2n}\]

Step 2: Simplify the series

Simplify the series using the fact that \((-1)^{n} \cdot (-1)^{n} = 1^{2n}=1\): \[g(x) = \sum_{n=0}^{\infty} x^{2n}\]

Step 3: Determine the interval of convergence

Since the original series converged for \(-1< x <1\), now, replacing \(x\) with \(-x^{2}\), we have \(-1 < -x^{2} < 1\). That leads to \(1 > x^{2} > -1\). Square rooting all components yield \(-1 < x < 1\). Thus, the interval of convergence remains the same. Note that the series does converge at \(x=\pm 1\), so the convergence interval is \(-1 \le x \le 1\).

Key Concepts

Interval of Convergence

In the context of power series, the interval of convergence is a very important concept. It tells us where the series converges, meaning where the infinite sum of the series approaches a certain finite number. Consider the series \( \sum_{n=0}^{\infty} x^{2n} \), which we can derive from replacing \(-x^2\) in another known series.

When determining the interval of convergence, one must identify the values of \( x \) for which the series converges. Originally, the given power series for \( \frac{1}{1+x} \) is known to converge for \(-1 < x < 1\). When we substitute \( x \) with \(-x^2\) we need to solve the inequality \(-1 < -x^2 < 1 \).

• This leads us to the inequality \(-1 < x^2 < 1\).
• Solving this, we find \(-1 < x < 1\), after considering the behavior around zero.
• We then must check the endpoints: \( x = \pm 1 \).

Because the series is even, convergence at these endpoints must be calculated separately. In this situation, the power series converges at \( x = -1 \) and \( x = 1 \), widening our interval of convergence to \([-1, 1]\). This extended interval indicates where the series reliably approximates the function.

Centered at 0

The phrase "centered at 0" or "around the origin" refers to how a series is structured in terms of its expansions. In power series, it is the value of \( x \) where the series "starts," or where the terms are built around. For instance, if we have a function with a power series representation \( g(x) = \sum_{n=0}^\infty c_n x^n \), it is centered at zero.

This centering can visibly affect how the series behaves or converges. It dictates how the powers of \( x \) are incremented, starting with \( x^0 \) and moving to higher powers like \( x^1, x^2, \) and so on.

• The convergence properties generally depend on this centering; here at 0, it simplifies calculations for convergence intervals.
• Centered at 0 means the series sums increase straightforwardly based on x's power.
• It also influences how we approach simplifying and substituting into other functions.

This positioning is particularly helpful when dealing with trigonometric or exponential functions common in calculus and ensures easy manipulation of polynomial terms, ensuring they begin from a unified point.

Series Convergence

A series is said to converge if the infinite sum of its terms approaches a specific, finite value. Understanding whether a series converges is crucial in mathematics, especially when dealing with power series, where each term is raised to increasing powers of \( x \).

To assess convergence, we need to consider tests and calculations:

• For geometric or similar series, their absolute sum should stay finite as we approach infinity.
• In our example, \( g(x) = \sum_{n=0}^{\infty} x^{2n} \), convergence is examined within given boundaries.
• Beyond absolute convergence, we also see if specifically termed convergently at certain integral endpoints.

It's critical, too, to differentiate between absolute convergence (convergent regardless of term sign) and conditional convergence (dependent on term arrangement).
For instance, our example shows convergence over \([-1,1]\) giving a solid base to approximate the underlying function accurately. This can be checked using the Ratio or Root tests, which consider limits based on series terms' subsequent behavior. Understanding when a series converges and why helps solidify using power series for functions that aren't otherwise easily assessable.