Radius Of Convergence R Of The Series ∑[n=1 To ∞] ((x-1)^n)/(n^2 * 3^n)
In the realm of mathematical analysis, power series hold a position of significant importance. These series, expressed in the form ∑[n=0 to ∞] a_n(x-c)^n, where a_n represents the coefficients, x is the variable, and c denotes the center of the series, are instrumental in approximating functions, solving differential equations, and modeling various physical phenomena. A crucial aspect of power series is their radius of convergence, which dictates the interval within which the series converges to a finite value. Understanding the concept of the radius of convergence is paramount for anyone delving into the world of power series and their applications.
Delving into Power Series and Convergence
Before we tackle the specific problem of finding the radius of convergence, let's first solidify our understanding of power series and convergence. A power series is essentially an infinite series where each term involves a power of (x-c), where 'c' is the center of the series. The behavior of a power series, particularly its convergence, depends heavily on the value of 'x'. A power series might converge for some values of 'x' and diverge for others. This brings us to the concept of the interval of convergence, which is the set of all 'x' values for which the series converges. The radius of convergence, denoted by 'R', is a non-negative real number that defines the size of this interval. It essentially tells us how far away from the center 'c' we can move before the series starts to diverge.
The radius of convergence (R) is the distance from the center 'c' to the nearest point where the series diverges. The interval of convergence is then given by (c-R, c+R), with the endpoints possibly included depending on the specific series. Determining the radius of convergence is a fundamental step in analyzing the behavior of a power series. Several methods exist for finding 'R', the most common being the ratio test and the root test. These tests provide a systematic way to analyze the convergence of a series by examining the ratio or root of consecutive terms.
Methods for Determining the Radius of Convergence
Two primary methods are used to determine the radius of convergence of a power series: the ratio test and the root test. These tests are powerful tools that allow us to analyze the convergence behavior of a series by examining the limit of certain expressions involving the series terms.
The Ratio Test
The ratio test is a widely used method for determining the convergence of a series. It involves calculating the limit of the ratio of consecutive terms in the series. Specifically, for a power series ∑[n=0 to ∞] a_n(x-c)^n, the ratio test considers the limit:
L = lim [n→∞] |a_(n+1)(x-c)^(n+1) / a_n(x-c)^n| = lim [n→∞] |a_(n+1) / a_n| |x-c|
If L < 1, the series converges absolutely. If L > 1, the series diverges. If L = 1, the test is inconclusive, and other methods are needed. The radius of convergence 'R' can be found by solving the inequality L < 1 for |x-c|. The resulting inequality will be of the form |x-c| < R, where R is the radius of convergence.
The Root Test
The root test provides an alternative approach to determining the convergence of a series. It involves calculating the limit of the nth root of the absolute value of the series terms. For a power series ∑[n=0 to ∞] a_n(x-c)^n, the root test considers the limit:
L = lim [n→∞] |a_n(x-c)n|(1/n) = lim [n→∞] |a_n|^(1/n) |x-c|
Similar to the ratio test, if L < 1, the series converges absolutely. If L > 1, the series diverges. If L = 1, the test is inconclusive. The radius of convergence 'R' can be found by solving the inequality L < 1 for |x-c|, which will again yield an inequality of the form |x-c| < R.
Applying the Ratio Test to the Given Series
Now, let's apply the ratio test to the given series: ∑[n=1 to ∞] ((x-1)n)/(n2 * 3^n). Here, a_n = 1/(n^2 * 3^n) and c = 1. We need to calculate the limit:
L = lim [n→∞] |a_(n+1)(x-1)^(n+1) / a_n(x-1)^n|
Substituting the expression for a_n, we get:
L = lim [n→∞] |(1/((n+1)^2 * 3(n+1)))(x-1)(n+1) / (1/(n^2 * 3n))(x-1)n|
Simplifying the expression, we have:
L = lim [n→∞] |(n^2 * 3^n) / ((n+1)^2 * 3^(n+1)) * (x-1)^(n+1) / (x-1)^n|
L = lim [n→∞] |(n^2 / (n+1)^2) * (3^n / 3^(n+1)) * (x-1)|
L = lim [n→∞] |(n^2 / (n^2 + 2n + 1)) * (1/3) * (x-1)|
As n approaches infinity, the term n^2 / (n^2 + 2n + 1) approaches 1. Therefore,
L = |(1) * (1/3) * (x-1)| = |(x-1)/3|
For the series to converge, we need L < 1. Thus,
|(x-1)/3| < 1
|x-1| < 3
This inequality tells us that the distance between x and the center 1 must be less than 3. Therefore, the radius of convergence R is 3.
Determining the Interval of Convergence
We've established that the radius of convergence R is 3 and the center c is 1. This means the interval of convergence is centered at 1 and extends 3 units in both directions. The interval is (1-3, 1+3) = (-2, 4). However, we need to check the endpoints x = -2 and x = 4 to see if the series converges at these points.
Checking x = -2
When x = -2, the series becomes:
∑[n=1 to ∞] ((-2-1)^n) / (n^2 * 3^n) = ∑[n=1 to ∞] ((-3)^n) / (n^2 * 3^n) = ∑[n=1 to ∞] ((-1)^n) / n^2
This is an alternating series, and the absolute value of the terms, 1/n^2, decreases monotonically to 0. By the alternating series test, this series converges.
Checking x = 4
When x = 4, the series becomes:
∑[n=1 to ∞] ((4-1)^n) / (n^2 * 3^n) = ∑[n=1 to ∞] (3^n) / (n^2 * 3^n) = ∑[n=1 to ∞] 1/n^2
This is a p-series with p = 2, and we know that p-series converge when p > 1. Therefore, this series converges.
Conclusion
Since the series converges at both endpoints, the interval of convergence includes the endpoints. The interval of convergence is [-2, 4]. However, the question specifically asks for the radius of convergence, which we have determined to be 3.
Final Answer
The radius of convergence R of the series ∑[n=1 to ∞] ((x-1)n)/(n2 * 3^n) is 3. Therefore, the correct answer is (c). This detailed exploration of the problem highlights the importance of understanding power series, convergence tests, and the concept of the radius of convergence. By applying the ratio test and analyzing the endpoints, we were able to confidently determine the radius of convergence for the given series.
