Problem 23 - A Course in Calculus and Real Analysis

Question

Find the radius of convergence of the power series \(\sum_{k=0}^{\infty} c_{k} x^{k}\) if for \(k \in \mathbb{N}\), the coefficient \(c_{k}\) is given as follows: (i) \(k !\), (ii) \(k^{2}\), (iii) \(\frac{k}{k^{2}+1}\), (iv) \(k e^{-k}\), (v) \(c^{k^{2}}\), where \(c \in \mathbb{R}\), (vi) \(\frac{k^{k}}{k !}\) (vii) \(\frac{2^{k}}{k^{2}}\) (viii) \(\left(\begin{array}{c}k+m \\ k\end{array}\right)\), where \(m \in \mathbb{N}\).

Step-by-Step Solution

Answer

(i) No radius of convergence, series does not converge for any value of \(x\). (ii) \(R = 1\), series converges for \(|x| < 1\). (iii) \(R = 1\), series converges for \(|x| < 1\). (iv) \(R = e\), series converges for \(|x| < e\). (v) \(R = |c|\), series converges for \(|x| < |c|\). (vi) \(R = \infty\), series converges for \(|x| = 0\). (vii) \(R = \frac{1}{2}\), series converges for \(|x| < \frac{1}{2}\). (viii) \(R = 1\), series converges for \(|x| < 1\).

Step 1: (i) Coefficient \(c_{k} = k !\)

We have the series \(\sum_{k=0}^{\infty} (k !) x^{k}\). To find the radius of convergence, apply the Ratio Test formula: \[ R=\lim_{k \to \infty} \sup | \frac{(k+1)!}{k !} | \] Notice that \((k+1)! = (k+1)k!\), so: \[ R = \lim_{k \to \infty} \sup | \frac{(k+1)k !}{k !} | \] The \(k!\) terms cancel out, leaving: \[ R = \lim_{k \to \infty} \sup |k+1| \] As \(k\) approaches infinity, the limit does not exist. Hence, there is no radius of convergence, and the series does not converge for any value of \(x\).

Step 2: (ii) Coefficient \(c_{k} = k^{2}\)

We have the series \(\sum_{k=0}^{\infty} (k^{2})x^{k}\). To find the radius of convergence, apply the Ratio Test formula: \[ R = \lim_{k \to \infty} \sup | \frac{(k+1)^{2}}{k^{2}} | \] Divide both terms of the fraction by \(k^{2}\) to get: \[ R = \lim_{k \to \infty} \sup | \frac{1 + 2k^{-1} + k^{-2}}{1} | \] As \(k\) approaches infinity, both \(2k^{-1}\) and \(k^{-2}\) go to zero, so the limit converges to 1. Thus, the radius of convergence is 1, and the series converges for \(|x| < 1\). Continue this same process for the rest of the coefficients:

Step 3: (iii) Coefficient \(c_{k} = \frac{k}{k^{2}+1}\)

R = \(\lim_{k \to \infty} \sup | \frac{(k+1)(k^{2}+1)}{k(k+1)^{2}+k(k+1)} | R = 1\), Series converges for \(|x| < 1\).

Step 4: (iv) Coefficient \(c_{k} = ke^{-k}\)

R = \(\lim_{k \to \infty} \sup | \frac{(k+1)e^{-(k+1)}}{ke^{-k}} | R = e\), Series converges for \(|x| < e\).

Step 5: (v) Coefficient \(c_{k} = c^{k^{2}}\)

R = \(\lim_{k \to \infty} \sup | \frac{c^{(k+1)^{2}}}{c^{k^{2}}} | R = |c|\), Series converges for \(|x| < |c|\).

Step 6: (vi) Coefficient \(c_{k} = \frac{k^{k}}{k !}\)

R = \(\lim_{k \to \infty} \sup | \frac{(k+1)^{(k+1)}}{k^{k}(k+1)} | R = \infty\), Series converges for \(|x| = 0\).

Step 7: (vii) Coefficient \(c_{k} = \frac{2^{k}}{k^{2}}\)

R = \(\lim_{k \to \infty} \sup | \frac{2^{(k+1)}(k^{2})}{2^{k}(k+1)^{2}} | R = \frac{1}{2}\), Series converges for \(|x| < \frac{1}{2}\).

Step 8: (viii) Coefficient \(c_{k} = \left(\begin{array}{c}k+m \\\ k\end{array}\right)\)

R = \(\lim_{k \to \infty} \sup | \frac{\left(\begin{array}{c}(k+1)+m \\\ (k+1)\end{array}\right)}{\left(\begin{array}{c}k+m \\\ k\end{array}\right)} | R = 1\), Series converges for \(|x| < 1\).

Key Concepts

Power Series

A power series is a type of infinite series that takes the form \(\sum_{k=0}^{\infty} a_k (x - c)^k\), where \(a_k\) are the coefficients and \(c\) is the center of the series. The series depends on a variable \(x\), and its behavior is influenced by the values of \(x\). This kind of series is quite powerful in mathematical analysis, commonly used for representing functions.

• Core Idea: Think of a power series as a polynomial with an infinite number of terms. Each term includes the variable \(x\) raised to a power \(k\), multiplied by a coefficient \(a_k\).
• Convergence: A power series may only converge—that is, produce a finite sum—for certain values of \(x\). These values typically fall within a certain range from the center \(c\), known as the interval of convergence.
• Radius of Convergence: The distance from the center \(c\) to the boundary of this interval is called the radius of convergence. It tells us how far around the center \(c\) we can safely use the series to represent functions or numbers.

Please keep in mind, the coefficients play a crucial role in determining both the interval and radius of convergence, often assessed using tests like the ratio test.

Ratio Test

The ratio test is an essential tool from mathematical analysis employed to determine the convergence of a series. To apply the ratio test for a power series, evaluate the limit of the absolute value of the ratio of consecutive terms.
The formula is typically given by:\[ R = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \]This value, \(R\), helps establish whether a series converges or not.

• If \(R < 1\): The series converges absolutely.
• If \(R > 1\): The series diverges.
• If \(R = 1\): The ratio test is inconclusive; other tests may be required.

When applied to power series, the ratio test specifically aids in determining the radius of convergence, \(R\). For cases such as \(c_k = k!\), the test indicates divergence as \(k\) approaches infinity, leading to no convergence for any \(x\). For other cases like \(c_k = k^2\), you may find \(R = 1\), indicating the series converges for \(|x|<1\).

Convergence

Convergence is a fundamental concept that examines whether an infinite series results in a finite sum. In applying to power series, convergence is tested to establish parameters for \(x\) where the series behaves regularly.

• Absolute Convergence: This occurs if the series of absolute values converges. With power series main interest lies in absolute convergence.
• Conditional Convergence: Less common in power series, occurs if a series converges but not absolutely.
• Testing Convergence: Important tests include the ratio test (commonly applied), root test, and others. Particularly for power series, ratio test is very useful to ascertain convergence and find the radius.

Through the example, we see that any power series only converges for \(x\) within a specific range, exploring these conditions allows constructing accurate function models over certain intervals.

Mathematical Analysis

Mathematical analysis involves the rigorous study of limits, functions, continuity, and infinite series like power series. It's a broad field aiming to provide deeper insights into the behavior of mathematical expressions under infinite conditions.

• Role in Series: Analysis allows us to rigorously verify where and why a series converges, drawing upon theorems and tests.
• Complexity: While seemingly challenging, it essentially revolves around understanding limits and applying tests to either confirm convergence or reveal divergence.
• Applications: This form of analysis is key in physics for series approximations and in engineering for signal processing, often involving power series.

Using the example provided, utilizing mathematical analysis and tests like the ratio test allows determination of convergence zones, enhancing understanding and application of series in various scientific fields.