Problem 69
Question
Let \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n},\) where \(c_{n+3}=c_{n}\) for \(n \geq 0\). (a) Find the interval of convergence of the series. (b) Find an explicit formula for \(f(x)\).
Step-by-Step Solution
Verified Answer
The interval of convergence for the given power series is \(|x| < 1\). The explicit formula for \(f(x)\) is \(f(x) = \frac{c_0}{1-x^3} + \frac{c_1x}{1-x^3} + \frac{c_2x^2}{1-x^3}\).
1Step 1: Breaking the infinite series into three separate series
The infinite series \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n}\) can be divided into three separate series to account for the coefficients \(c_{n+3}=c_{n}\). Hence, we write it as follows: \(f(x) = c_{0}\sum_{n=0}^{\infty} (x^3)^n + c_1\sum_{n=0}^{\infty} x(x^3)^n + c_2\sum_{n=0}^{\infty} x^2(x^3)^n\)
2Step 2: Application of the ratio test to find the interval of convergence
By applying the ratio test to each of these series, it can be determined that the interval of convergence for all three series is \(|x| > 1\). Consequently, the interval of convergence for the original series is the same, \(|x| < 1\).
3Step 3: Finding explicit formula for \(f(x)\)
To find the explicit formula for \(f(x)\), apply formula for sum of an infinite geometric series to each of the three series. This yields \(f(x) = \frac{c_0}{1-x^3} + \frac{c_1x}{1-x^3} + \frac{c_2x^2}{1-x^3}\).
Key Concepts
Interval of ConvergenceRatio TestInfinite SeriesGeometric Series
Interval of Convergence
When discussing power series, one key concept is the "interval of convergence." This term refers to the set of all real numbers where the series converges. Simply put, it's the range of values you can plug into the series without making it blow up to infinity.
To find this interval, techniques such as the **ratio test** are used, which will be discussed in detail later.
In our example, breaking the infinite series into three smaller series showed that each follows the same pattern. The interval result, \( |x| < 1 \), indicates where the series safely converges. This interval is a crucial detail in understanding how functions like our given series behave.
Knowing where a series converges can help you figure out where to apply the function and predict its behavior in different scenarios.
To find this interval, techniques such as the **ratio test** are used, which will be discussed in detail later.
In our example, breaking the infinite series into three smaller series showed that each follows the same pattern. The interval result, \( |x| < 1 \), indicates where the series safely converges. This interval is a crucial detail in understanding how functions like our given series behave.
Knowing where a series converges can help you figure out where to apply the function and predict its behavior in different scenarios.
Ratio Test
The ratio test is a powerful tool used to determine convergence or divergence of an infinite series. It evaluates the limit of the absolute value of the ratio of consecutive terms.
For a series \(\sum a_n\), it looks at \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). If the result is less than 1, the series converges absolutely. If it's more than 1, the series diverges.
In the context of our exercise, applying the ratio test to each of the three broken-down series showed that they all converge for \( |x| < 1 \). This is critical since knowing when a series behaves well can save you from errors in calculations.
For a series \(\sum a_n\), it looks at \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\). If the result is less than 1, the series converges absolutely. If it's more than 1, the series diverges.
In the context of our exercise, applying the ratio test to each of the three broken-down series showed that they all converge for \( |x| < 1 \). This is critical since knowing when a series behaves well can save you from errors in calculations.
Infinite Series
An infinite series is simply the sum of an infinite sequence of numbers. In notation, it's typically written as \(\sum_{n=0}^{\infty} a_n\).
Understanding infinite series is foundational for topics like calculus and real analysis, as it helps approximate functions and solve complex problems.
In our exercise, an infinite series is expressed as \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n} \). Given the condition \(c_{n+3}=c_{n}\), the series has a periodic coefficient pattern, which allows it to be divided neatly into three smaller series. This shows the flexibility and adaptability of infinite series in representing functions.
Understanding infinite series is foundational for topics like calculus and real analysis, as it helps approximate functions and solve complex problems.
In our exercise, an infinite series is expressed as \(f(x)=\sum_{n=0}^{\infty} c_{n} x^{n} \). Given the condition \(c_{n+3}=c_{n}\), the series has a periodic coefficient pattern, which allows it to be divided neatly into three smaller series. This shows the flexibility and adaptability of infinite series in representing functions.
Geometric Series
A geometric series is a specific type of series where each term is found by multiplying the previous term by a constant called the ratio. It's formatted as \(a + ar + ar^2 + \ldots\).
The sum of an infinite geometric series is particularly simple: \(\frac{a}{1-r}\), where \(|r| < 1\).
In the given exercise, the series are essentially geometric since the terms are repeated with the same ratios. By recognizing this, we were able to use the formula for the sum of a geometric series to find an explicit formula for \(f(x) \). Each piece of the function was expressed by using this neat trick, turning a complex series into a simple, manageable formula.
The sum of an infinite geometric series is particularly simple: \(\frac{a}{1-r}\), where \(|r| < 1\).
In the given exercise, the series are essentially geometric since the terms are repeated with the same ratios. By recognizing this, we were able to use the formula for the sum of a geometric series to find an explicit formula for \(f(x) \). Each piece of the function was expressed by using this neat trick, turning a complex series into a simple, manageable formula.
Other exercises in this chapter
Problem 69
State the definitions of convergent and divergent series.
View solution Problem 69
Determine whether the sequence with th given \(n\) th term is monotonic. Discuss the boundedness of th sequence. Use a graphing utility to confirm your results.
View solution Problem 69
Test for convergence or divergence, using each test at least once. Identify which test was used. (a) \(n\) th-Term Test (b) Geometric Series Test (c) \(p\) -Ser
View solution Problem 69
Prove that \(\lim _{n \rightarrow \infty} \frac{x^{n}}{n !}=0\) for any real \(x\).
View solution