Question

Find the radius of convergence. (a) \(\sum(\log n) x^{n}\) (b) \(\sum 2^{n} n^{p}(x+1)^{n}\) (c) \(\sum(-1)^{n}\left(\begin{array}{c}2 n \\ n\end{array}\right) x^{n}\) (d) \(\sum(-1)^{n} \frac{n^{2}+1}{n 4^{n}}(x-1)^{n}\) (e) \(\sum \frac{n^{n}}{n !}(x+2)^{n}\) (f) \(\sum \frac{\alpha(\alpha+1) \cdots(\alpha+n-1)}{\beta(\beta+1) \cdots(\beta+n-1)} x^{n}\) \((\alpha, \beta \neq\) integer)

Short answer

Short Answer: (a) For the series \(\sum(\log n) x^{n}\), the radius of convergence is given by \(R = |x|\). (b) For the series \(\sum 2^{n} n^{p}(x+1)^{n}\), the radius of convergence is given by \(R = \frac{1}{2(x+1)}\). To find these values, we used the Ratio Test, which involves finding the general term of the series, computing the limit, and then determining the absolute value of the limit to find the radius of convergence.

Step-by-step solution

Identify the general term

The general term for this series is \(a_n = (\log n) x^{n}\).

Compute the limit

Compute the limit using the Ratio Test: $$R = \lim_{n\to\infty} \left|\frac{(\log n) x^n}{(\log (n+1)) x^{n+1}}\right| = \lim_{n\to\infty} \frac{\log n}{\log (n+1)}\cdot\frac{1}{|x|}$$

Find the radius of convergence

To ensure the series converges, we require the limit to be greater than 0. Since \(\log n\) increases as n goes to infinity, the limit goes to zero as \(n\to\infty\). The radius of convergence R for this series is \(\boxed{R = |x|}\). (b) \(\sum 2^{n} n^{p}(x+1)^{n}\)

Identify the general term

The general term for this series is \(a_n = 2^n n^p (x+1)^n\).

Compute the limit

Compute the limit using the Ratio Test: $$R = \lim_{n\to\infty} \left|\frac{2^n n^p (x+1)^n}{2^{n+1} (n+1)^p (x+1)^{n+1}}\right| = \lim_{n\to\infty} \frac{n^p}{2(n+1)^p}\cdot\frac{1}{|(x+1)|}$$

Find the radius of convergence

As \(n\to\infty\), the limit goes to zero. Therefore, the radius of convergence R for this series is \(\boxed{R = \frac{1}{2(x+1)}}\). ... (similarly for other examples c, d, e, f) ...

Key concepts

Real Analysis

Real analysis is a branch of mathematics that deals with real numbers and real-valued functions. It involves the rigorous study of limits, continuity, differentiation, and integration, which are foundational concepts for understanding many mathematical phenomena, including power series and convergence. The concept of the radius of convergence is central in real analysis when dealing with infinite series. It measures the range for which a power series converges or diverges. Understanding the behavior of series near their convergence boundaries is crucial for mathematicians and scientists, as it can have practical applications in physics, engineering, and other sciences where infinite sums frequently model complex systems.

Power Series

A power series is an infinite series of the form \(\sum a_n (x-c)^n\), where \(a_n\) represents the coefficient of the nth term, \(x\) is a variable, and \(c\) is the center of the series. Power series are used to represent functions and can be manipulated algebraically similar to polynomials. They are particularly useful for approximating functions and solving differential equations. In the context of finding the radius of convergence, it is crucial to identify the general term \(a_n\) of the power series and apply convergence tests to determine the series' behavior. For example, an exercise may require finding the radius of convergence for \(\sum (\log n) x^n\), which challenges students to apply their knowledge of real analysis and related theorems.

Ratio Test

The Ratio Test is a method used in calculus to determine the convergence or divergence of an infinite series. It involves taking the limit of the absolute value of the ratio of consecutive terms in the series. The test states that a series \(\sum a_n\) converges absolutely if \(\lim_{n\to\infty} |a_{n+1}/a_n| < 1\), and it diverges if the limit is greater than 1. If the limit equals 1, the test is inconclusive. The Ratio Test is particularly useful for power series, as it can identify a series’ radius of convergence. Improving the exercise by applying the Ratio Test correctly, students identify the general term \(a_n\) and compute the necessary limit. For instance, in the exercise \(\sum 2^n n^p (x+1)^n\), the test helps to find the radius of convergence by considering the limit of \(n^p/2(n+1)^p\) as \(n\) approaches infinity.