Question

Explain why a power series is tested for absolute convergence.

Short answer

Answer: It is important to test a power series for absolute convergence because this determines the interval of convergence, which is the range of values of x for which the power series converges. When a power series converges absolutely, it is numerically stable and can be manipulated more freely without affecting convergence. The main method used for testing absolute convergence is the Ratio Test, which involves calculating a limit L and determining the values of x for which L < 1.

Step-by-step solution

Definition of a power series

A power series is a series of the form ∑(c_n * (x-a)^n) where n is a non-negative integer, c_n is a coefficient, x is a variable, and a is a constant.

Absolute convergence

A series ∑a_n is said to be absolutely convergent if the series of the absolute values of its terms, ∑|a_n|, converges as well. It is known that if a series is absolutely convergent, it is also convergent. But, the converse is not necessarily true.

Testing for absolute convergence using Ratio Test

One of the most common tests for absolute convergence of a power series is the Ratio Test. It states that, given a series ∑a_n, let L = lim (n→∞) (|a_(n+1)| / |a_n|). If L < 1, the series converges absolutely. If L > 1, the series diverges, or if L = 1, the test is inconclusive.

Applying the Ratio Test to a power series

Given a power series ∑(c_n * (x-a)^n), we apply the Ratio Test by considering the series ∑|c_n * (x-a)^n|. We calculate the limit L = lim (n→∞) (|c_(n+1) * (x-a)^(n+1)| / |c_n * (x-a)^n|). Then, we determine the values of x for which L < 1, assuring absolute convergence.

Interval and radius of convergence

The interval of convergence (IOC) is the set of all values of x for which a given power series converges absolutely. The radius of convergence (ROC) is half the width of the interval of convergence. Typically, the IOC can be given in the form (a-R, a+R), where R is the radius of convergence. To find the endpoints of IOC, plug in x=a-R and x=a+R and analyze the convergence on these points.

Importance of absolute convergence

When a power series converges absolutely in its interval of convergence, it is numerically stable, and hence, can be manipulated more easily. Absolute convergence also allows for different operations such as termwise differentiation and integration, termwise limit, and rearrangement of order of the terms in the series without affecting the convergence.

Key concepts

Absolute Convergence

When we talk about absolute convergence, we are considering a series where if we take the absolute value of each term and the new series still converges, then the original series is considered absolutely convergent. This notion is very important for power series because if a series is absolutely convergent, it guarantees the convergence of the original series as well. Though the opposite is not true; a series might converge without being absolutely convergent.

- **Why is Absolute Convergence Important?** - Provides a flexible framework for function manipulation. - Ensures certain mathematical operations like differentiation and integration can safely be applied term by term.

By ensuring absolute convergence, we're able to trust the series to behave reliably within its interval of convergence, allowing us to rearrange terms without diverging.

Ratio Test

The Ratio Test is a powerful tool for determining whether a power series is absolutely convergent. It is particularly useful because it simplifies the process of analyzing series and offers a clear-cut criterion.

To apply the Ratio Test:- Write down the series as ∑a_n.- Calculate L: \( L = \lim_{{n \to \infty}} \frac{|a_{n+1}|}{|a_n|} \).- If \( L < 1 \), the series converges absolutely.- If \( L > 1 \), the series diverges.- If \( L = 1 \), the test is inconclusive, and you may need another test.

When using the Ratio Test for a power series, remember to apply it to the absolute values. It helps us identify the values of 'x' within which our series is stable and convergent.

Interval of Convergence

A power series can potentially converge or diverge based on the value of 'x'. The interval of convergence specifies the range of x values for which the series converges absolutely. This is essential for understanding the behavior and applicability of the series.

To find this interval: - Use tests like the Ratio Test and solve for x where the series is absolutely convergent. - The solution usually forms the interval (a-R, a+R), where 'R' is the radius of convergence, and 'a' is the center of the series.

The interval may include endpoints or not, depending on whether the series converges at those points, requiring further checks. Ensuring series convergence over an entire interval allows functions represented by power series to be effectively used in applied and theoretical contexts.

Radius of Convergence

The radius of convergence is a measure important for understanding how wide the interval of convergence is. It describes how far x can stray from 'a' while still converging absolutely. - **How to Find the Radius:** - Often, R can be derived from applying the Ratio Test. - Use the limits of the test to identify R as part of determining the interval.

Given as R in the interval (a-R, a+R), it is half the distance across the interval of convergence.

A large radius implies a robust series able to converge over a wide range, offering flexibility in function representation and approximations, whereas a smaller radius confines practical applications within a narrow scope. Exploring these concepts helps to strengthen comprehension of how and where a power series accurately and efficiently performs.