91Ó°ÊÓ

When working with power series, finding the interval of convergence is a key task. The interval of convergence is the range of values for which a power series converges. Every power series has an associated radius of convergence, denoted by \( R \).
If \( R \) is infinite, the series converges for all real numbers, resulting in an interval of convergence from \( -\infty \) to \( \infty \). If \( R \) is finite, then the interval of convergence is \((c - R, c + R)\), where \( c \) is the center of the series.

• Calculate \( R \) using the ratio or root test.
• Consider endpoints separately as they might converge or diverge.
• Express the interval with inequalities or interval notation.

In the given exercise, since \( R = \infty \), the series converges for all values of \( x \), leading to an interval of convergence of \((-\infty, \infty)\). This means that no matter what value you choose for \( x \), the series will sum to a finite number.

A power series is a type of series that takes the form \( \sum_{k=0}^{\infty} c_k x^k \). This series essentially represents a polynomial with infinitely many terms centered around a particular point, often at \( x = 0 \), making it a Maclaurin series.
The coefficients \( c_k \) play a crucial role in determining the behavior and convergence of the series. These values dictate how rapidly the terms decrease, impacting convergence.

• A power series resembles a finite polynomial but contains terms of all powers of \( x \).
• It can represent various functions, especially when converging over a particular interval.
• The nature and form of \( c_k \) influence the series' convergence.

In this exercise, \( c_k = \frac{3^k}{k!} \), exhibiting exponential behavior due to the factorial in the denominator. This particular form is related closely to the exponential function.

The ratio test is a technique used to determine the convergence of a series, especially efficient for power series. It involves calculating the limit of the absolute ratio of consecutive terms. Formally, it states that a series \( \sum a_k \) converges absolutely if\[ L = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1 \]
If \( L > 1 \), the series diverges. If \( L = 1 \), the test is inconclusive. The ratio test helps identify the radius of convergence \( R \) by finding \( R = \frac{1}{L} \) when \( L \) is between 0 and 1.

• Calculate \( L \) using the precise formula and simplify as needed.
• Check if \( L \) is less than 1 for convergence.
• Recognize that \( \frac{1}{L} \) determines \( R \), unless \( L = 0 \) indicating infinite convergence.

In the original problem, the ratio \( L \) is 0, indicating convergence for all \( x \), hence \( R = \infty \). This application of the ratio test simplifies understanding convergence behavior effectively.