Chapter 8: Problem 19
A power series is given. (a) Find the radius of convergence. (b) Find the interval of convergence. $$\sum_{n=0}^{\infty} \frac{3^{n}}{n !}(x-5)^{n}$$
Short Answer
Expert verified
(a) \(R = \infty\); (b) Interval of convergence is \((-\infty, \infty)\).
Step by step solution
01
Identify the Form of the Power Series
The given power series is \(\sum_{n=0}^{\infty} \frac{3^{n}}{n !}(x-5)^{n}\). This is a power series centered at \(x = 5\) with general term \(a_n = \frac{3^n}{n!}\cdot (x-5)^n\).
02
Use the Ratio Test for Convergence
To find the radius of convergence, use the ratio test. Calculate the limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]where:\[a_{n+1} = \frac{3^{n+1}}{(n+1)!} (x-5)^{n+1}\]\[a_{n} = \frac{3^n}{n!} (x-5)^{n}\]Thus,\[L = \lim_{n \to \infty} \left| \frac{3^{n+1}}{(n+1)!} \cdot \frac{n!}{3^n} \right| \cdot |x-5| = \lim_{n \to \infty} \frac{3|x-5|}{n+1}\]
03
Evaluate the Limit
The limit obtained is:\[L = \lim_{n \to \infty} \frac{3|x-5|}{n+1}\]As \(n \to \infty\), the limit \(L = 0\) for any value of \(x\), because \(\frac{1}{n+1} \to 0\). Therefore, the series converges for all \(x\).
04
Determine the Radius of Convergence
Since the series converges for all values of \(x\), the radius of convergence \(R\) is \(\infty\).
05
Determine the Interval of Convergence
Since the radius of convergence is infinity, the interval of convergence is the entire set of real numbers, which is \((-\infty, \infty)\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Radius of Convergence
In the world of power series, the radius of convergence is a crucial concept. It determines the range of values for which the series converges to a finite sum. Essentially, it tells us the 'distance' from the center of the series within which convergence is guaranteed.
For the power series given in the exercise, \[\sum_{n=0}^{\infty} \frac{3^{n}}{n !}(x-5)^{n},\]we apply the ratio test to find this radius. The radius of convergence is denoted by \( R \). If \( R \) is finite, the series converges for \( |x-c| < R \), where \( c \) is the center of the series. However, if \( R \) is infinite, as it is in this case, the series converges for all real numbers \( x \). This infinite radius implies no restrictions on \( x \), indicating universal convergence.
For the power series given in the exercise, \[\sum_{n=0}^{\infty} \frac{3^{n}}{n !}(x-5)^{n},\]we apply the ratio test to find this radius. The radius of convergence is denoted by \( R \). If \( R \) is finite, the series converges for \( |x-c| < R \), where \( c \) is the center of the series. However, if \( R \) is infinite, as it is in this case, the series converges for all real numbers \( x \). This infinite radius implies no restrictions on \( x \), indicating universal convergence.
Interval of Convergence
Once the radius of convergence is determined, the interval of convergence follows directly from it. The interval of convergence specifies the exact range of \( x \) values for which the power series converges.
In our given power series, since the radius of convergence \( R \) is infinite, the interval of convergence encompasses all real numbers. This means that for every real \( x \), the series converges nicely.
In mathematical terms, the interval is denoted as \((-\infty, \infty)\). It is important to note that if the radius were finite, we would also need to check the endpoints of the interval separately to confirm convergence at those specific points.
In our given power series, since the radius of convergence \( R \) is infinite, the interval of convergence encompasses all real numbers. This means that for every real \( x \), the series converges nicely.
In mathematical terms, the interval is denoted as \((-\infty, \infty)\). It is important to note that if the radius were finite, we would also need to check the endpoints of the interval separately to confirm convergence at those specific points.
Ratio Test
The ratio test is a powerful tool to determine the convergence of series, including power series. It involves examining the ratio of consecutive terms in a sequence. The test is particularly useful for identifying the radius of convergence.
With our series, using the ratio test involves calculating the following limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]where \(a_n\) is the general term of our power series. For this exercise, after simplification, we get:\[L = \lim_{n \to \infty} \frac{3|x-5|}{n+1}\]Since \(L\) approaches 0 as \( n \to \infty \), the ratio test confirms that the series converges for all \( x \), resulting in an infinite radius of convergence.
With our series, using the ratio test involves calculating the following limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]where \(a_n\) is the general term of our power series. For this exercise, after simplification, we get:\[L = \lim_{n \to \infty} \frac{3|x-5|}{n+1}\]Since \(L\) approaches 0 as \( n \to \infty \), the ratio test confirms that the series converges for all \( x \), resulting in an infinite radius of convergence.
Convergence
Convergence is a fundamental concept when dealing with power series and infinite series in general. It describes whether a series approaches a specific value as more and more terms are added.
For the power series in question, convergence needs to be checked for different values of \( x \). The use of the ratio test has shown that the given series converges for any real \( x \).
This convergence behavior indicates that no matter what \( x \) value you choose, the infinite sum of the series will settle towards a specific value, which is key in applications such as calculus and solving differential equations.
For the power series in question, convergence needs to be checked for different values of \( x \). The use of the ratio test has shown that the given series converges for any real \( x \).
This convergence behavior indicates that no matter what \( x \) value you choose, the infinite sum of the series will settle towards a specific value, which is key in applications such as calculus and solving differential equations.
Infinite Series
An infinite series is simply the sum of countlessly many terms, which can be daunting at first. They are a central topic in advanced mathematics, with applications ranging from calculus to complex analysis.
In the context of our problem, the power series:\[\sum_{n=0}^{\infty} \frac{3^{n}}{n !}(x-5)^{n}\]is a specific type of infinite series where each term includes a power of \((x-5)\).
Infinite series can converge to a limit, allowing us to make sense of the infinite sum in a finite way. Understanding the conditions under which they converge is where the radius and interval of convergence come into play, alongside tests like the ratio test.
In the context of our problem, the power series:\[\sum_{n=0}^{\infty} \frac{3^{n}}{n !}(x-5)^{n}\]is a specific type of infinite series where each term includes a power of \((x-5)\).
Infinite series can converge to a limit, allowing us to make sense of the infinite sum in a finite way. Understanding the conditions under which they converge is where the radius and interval of convergence come into play, alongside tests like the ratio test.
