Convergence Analysis Of The Infinite Series ∑(n=1 To ∞) ((n+1)² / 2ⁿ) (x-3)ⁿ

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Introduction to Infinite Series

In the vast landscape of mathematics, the concept of infinite series holds a significant place, bridging the gap between finite sums and the infinite. These series, represented as the sum of an infinite number of terms, are fundamental in various areas, including calculus, real analysis, and complex analysis. Understanding infinite series is crucial for solving complex problems and modeling real-world phenomena. Our focus in this article is the in-depth exploration of the specific infinite series: ${ \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (x-3)^n }$ This series presents a unique challenge and opportunity to delve into the intricacies of convergence, divergence, and the behavior of series in relation to the variable x. In the following sections, we will dissect this series, examining its components and employing various mathematical tools to determine its properties. We will begin by defining the series and discussing its general form. Then, we will analyze the convergence of the series using the ratio test and root test, which are powerful methods for determining the radius and interval of convergence. Furthermore, we will explore the endpoints of the interval of convergence to ascertain whether the series converges or diverges at these critical points. By understanding these aspects, we can gain a comprehensive understanding of the given infinite series and its applications in various mathematical contexts. The journey through this series will not only enhance our knowledge of series but also strengthen our analytical and problem-solving skills in mathematics.

Defining the Infinite Series and its Components

To truly grasp the essence of an infinite series like ${ \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (x-3)^n, }$ we must first dissect its components and understand how they interact. This particular series is a power series, a special type of infinite series where each term involves a power of the variable x. In this case, the series is centered around x = 3, as evidenced by the (x-3)^n term. This centering is a crucial aspect of power series, as it determines the point around which the series' behavior is analyzed. The general form of a power series centered at a is given by:

${ \sum_{n=0}^{\infty} c_n (x-a)^n, }$

where c_n represents the coefficients of the series. Comparing this general form with our given series, we can identify the coefficients as:

${ c_n = \frac{(n+1)^2}{2^n} }$

These coefficients play a vital role in determining the convergence and behavior of the series. The (n+1)^2 term in the numerator contributes to the growth of the terms as n increases, while the 2^n term in the denominator acts to dampen this growth. The interplay between these terms is critical in determining whether the series converges or diverges. The (x-3)^n term introduces the variable x, making the convergence of the series dependent on the value of x. Specifically, the series will converge for certain values of x and diverge for others. The set of x values for which the series converges forms the interval of convergence, a fundamental concept in the analysis of power series. The radius of convergence, denoted by R, is a measure of the size of this interval. Understanding the radius and interval of convergence is essential for determining the domain over which the series provides a meaningful representation of a function. In the subsequent sections, we will delve deeper into the methods for finding the radius and interval of convergence for our given series, employing techniques such as the ratio test and root test. This exploration will provide a comprehensive understanding of the series' behavior and its applicability in various mathematical contexts.

Convergence Tests: Ratio Test and Root Test

Determining the convergence of an infinite series is a fundamental aspect of its analysis. For the series ${ \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (x-3)^n, }$ we employ powerful tools such as the ratio test and the root test to establish its convergence properties. The ratio test is particularly effective for series where the terms involve factorials or exponential functions, while the root test is useful when the terms have powers. Both tests provide a criterion for convergence based on the limit of a certain ratio involving consecutive terms of the series.

The ratio test states that for a series ∑a_n, if the limit

${ L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| }$

exists, then:

• If L < 1, the series converges absolutely.
• If L > 1, the series diverges.
• If L = 1, the test is inconclusive.

Applying the ratio test to our series, we have:

${ a_n = \frac{(n+1)^2}{2^n} (x-3)^n }$

and

${ a_{n+1} = \frac{(n+2)^2}{2^{n+1}} (x-3)^{n+1} }$

Thus,

${ \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{\frac{(n+2)^2}{2^{n+1}} (x-3)^{n+1}}{\frac{(n+1)^2}{2^n} (x-3)^n} \right| = \frac{(n+2)^2}{2(n+1)^2} |x-3| }$

Taking the limit as n approaches infinity:

${ L = \lim_{n \to \infty} \frac{(n+2)^2}{2(n+1)^2} |x-3| = \frac{|x-3|}{2} \lim_{n \to \infty} \frac{(n+2)^2}{(n+1)^2} = \frac{|x-3|}{2} }$

For convergence, we require L < 1, which gives:

${ \frac{|x-3|}{2} < 1 \implies |x-3| < 2 }$

This inequality implies that the series converges for x values within a distance of 2 from the center x = 3. Therefore, the radius of convergence R is 2. The interval of convergence is (3 - 2, 3 + 2) = (1, 5). However, we must also check the endpoints x = 1 and x = 5 to determine the complete interval of convergence.

Alternatively, we can use the root test, which states that for a series ∑a_n, if the limit

${ L = \lim_{n \to \infty} \sqrt[n]{|a_n|} }$

exists, then:

• If L < 1, the series converges absolutely.
• If L > 1, the series diverges.
• If L = 1, the test is inconclusive.

Applying the root test to our series:

${ \sqrt[n]{|a_n|} = \sqrt[n]{\left| \frac{(n+1)^2}{2^n} (x-3)^n \right|} = \frac{\sqrt[n]{(n+1)^2}}{2} |x-3| }$

Taking the limit as n approaches infinity:

${ L = \lim_{n \to \infty} \frac{\sqrt[n]{(n+1)^2}}{2} |x-3| = \frac{|x-3|}{2} \lim_{n \to \infty} (n+1)^{\frac{2}{n}} }$

Since ${\lim_{n \to \infty} (n+1)^{\frac{2}{n}} = 1}$, we have:

${ L = \frac{|x-3|}{2} }$

This leads to the same condition for convergence as the ratio test, |x - 3| < 2, confirming the radius of convergence R = 2 and the interval of convergence (1, 5). In the next section, we will examine the behavior of the series at the endpoints of this interval to fully characterize its convergence.

Endpoint Analysis: Determining Convergence at x = 1 and x = 5

After establishing the radius of convergence and the open interval of convergence for the infinite series ${ \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (x-3)^n, }$ we must now analyze the endpoints of the interval to determine the complete interval of convergence. The endpoints are the values of x where the convergence tests (ratio and root tests) are inconclusive, and thus require separate investigation. In our case, the open interval of convergence is (1, 5), so we need to examine the series' behavior at x = 1 and x = 5.

Case 1: x = 1

Substituting x = 1 into the series, we obtain:

${ \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (1-3)^n = \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (-2)^n = \sum_{n=1}^{\infty} (-1)^n (n+1)^2 }$

This is an alternating series, but the absolute value of the terms, (n+1)^2, increases without bound as n approaches infinity. Therefore, the terms do not approach zero, violating the necessary condition for convergence of a series. Specifically, the limit

${ \lim_{n \to \infty} (-1)^n (n+1)^2 }$

does not exist. Consequently, the series diverges at x = 1.

Case 2: x = 5

Substituting x = 5 into the series, we obtain:

${ \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (5-3)^n = \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (2)^n = \sum_{n=1}^{\infty} (n+1)^2 }$

This series is a positive term series, where each term is (n+1)^2. As n approaches infinity, the terms (n+1)^2 also approach infinity. Thus, the limit

${ \lim_{n \to \infty} (n+1)^2 = \infty }$

Since the terms do not approach zero, the series diverges at x = 5. This can also be seen by recognizing that the terms of the series grow quadratically, and thus the sum will grow without bound.

Conclusion of Endpoint Analysis

From our analysis, we have found that the series diverges at both x = 1 and x = 5. Therefore, the endpoints are not included in the interval of convergence. Combining this with the open interval of convergence (1, 5) obtained from the ratio and root tests, we conclude that the complete interval of convergence for the given series is (1, 5). This means that the series converges for all x values strictly between 1 and 5, and diverges for all other x values. Understanding the behavior of a power series at its endpoints is crucial for defining the domain over which the series can be used to represent a function or solve mathematical problems. In the next section, we will summarize our findings and discuss the implications of the interval of convergence for the given series.

Summary and Conclusion

In this comprehensive exploration, we have thoroughly analyzed the infinite series ${ \sum_{n=1}^{\infty} \frac{(n+1)^2}{2^n} (x-3)^n. }$ Our investigation began with a detailed examination of the series' components, recognizing it as a power series centered at x = 3. We identified the coefficients and the role of the (x-3)^n term in determining the series' behavior based on the value of x.

To determine the convergence properties of the series, we employed the ratio test and the root test, two powerful tools for analyzing infinite series. Both tests led us to the same conclusion: the radius of convergence R is 2, and the open interval of convergence is (1, 5). This means that the series converges for all x values within this interval.

However, the analysis did not end there. We recognized the importance of examining the endpoints of the interval, x = 1 and x = 5, to determine the complete interval of convergence. By substituting these values into the series, we found that the series diverges at both endpoints. At x = 1, the series becomes an alternating series whose terms do not approach zero, violating the necessary condition for convergence. At x = 5, the series becomes a positive term series whose terms also do not approach zero, leading to divergence.

Therefore, our final conclusion is that the complete interval of convergence for the given series is (1, 5), excluding the endpoints. This interval defines the set of x values for which the series converges to a finite sum. Outside this interval, the series diverges, meaning it does not have a finite sum.

The significance of this result lies in its implications for the applications of the series. Within the interval of convergence, the series can be used to represent a function, allowing us to approximate its values and perform calculus operations such as differentiation and integration. Outside the interval, the series representation is not valid. Understanding the interval of convergence is therefore crucial for the correct and meaningful use of power series in mathematical analysis and problem-solving.

In summary, we have successfully determined the interval of convergence for the given infinite series by employing a combination of convergence tests and endpoint analysis. This process highlights the importance of a thorough and rigorous approach to the analysis of infinite series, ensuring a complete understanding of their behavior and applicability.