Problem 48 - Calculus: One and Several Variables

Question

Let \(\sum a_{k} x^{k}\) be a power series with finite radius of convergence \(r\). Show that the power series \(\sum a_{k} x^{2 k}\) has radius of convergence \(\sqrt{r}\)

Step-by-Step Solution

Answer

The short version of the answer is as follows: First, apply the ratio test to the power series \(\sum a_{k} x^{k}\) and use the definition of radius of convergence to find the radius of convergence, \(r\). Then, apply the ratio test to the power series \(\sum a_{k} x^{2 k}\) and find the radius of convergence for this power series as well. By comparing the results, we can show that the radius of convergence for the power series \(\sum a_{k} x^{2 k}\) is \(\sqrt{r}\).

Step 1: Recall the Ratio Test and the definition of radius of convergence.

The ratio test states that if the limit of the absolute value of the ratio of consecutive terms of a series is a constant C: \[\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = C \] for a series to converge, C must be smaller than 1, and for a series to diverge, C must be greater than 1. The radius of convergence R for a power series is the radius of the interval around 0 for which the power series converges. That is, the power series converges for \(|x| < R\).

Step 2: Find the radius of convergence for the power series \(\sum a_{k} x^{k}\)

Apply the ratio test to the power series: \[\lim_{k \to \infty} \left| \frac{a_{k+1} x^{k+1}}{a_k x^k} \right| = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \cdot |x| = C\] The power series converges when \(C < 1\). Therefore, for convergence: \[\left| \frac{a_{k+1}}{a_k} \right| \cdot |x| < 1\] Given that the radius of convergence is \(r\), this implies that: \[|x| < \frac{1}{\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|} = r\]

Step 3: Apply the Ratio Test to the power series \(\sum a_{k} x^{2 k}\)

The power series is now \(\sum a_{k} x^{2 k}\). Apply the ratio test: \[\lim_{k \to \infty} \left| \frac{a_{k+1} x^{2(k+1)}}{a_k x^{2k}} \right| = \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \cdot |x^2| = C'\] To have convergence, \(C' < 1\). Therefore: \[\left| \frac{a_{k+1}}{a_k} \right| \cdot |x^2| < 1\]

Step 4: Find the radius of convergence for the power series \(\sum a_{k} x^{2 k}\)

We know that the radius of convergence for the original power series is \(r\). Use this information to find the radius of convergence for the modified power series: \[\left| \frac{a_{k+1}}{a_k} \right| \cdot |x^2| < 1\] \[|x^2| < \frac{1}{\lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right|} = r\] Take the square root of both sides to obtain the radius of convergence for the new power series: \[|x| < \sqrt{r}\] Therefore, the radius of convergence for the power series \(\sum a_{k} x^{2 k}\) is \(\sqrt{r}\).

Key Concepts

Power Series

A power series is a way to express functions as infinite sums of terms involving powers of a variable. It takes the form \[ \sum_{k=0}^{\infty} a_k x^k \]where \(a_k\) are coefficients and \(x\) is the variable. Each term increases the power of \(x\), which makes power series very flexible in representing a wide range of functions. With the right coefficients, functions like exponentials, sines, and polynomials can be represented this way.

Power series are centered around a particular value, often zero, and can represent a function within a certain radius from this center, known as the "radius of convergence." This radius determines how far from the center you can go while the series still accurately represents the function. If \( \sum a_{k} x^{k} \) has a radius of convergence \(r\), it converges for all \(|x| < r\). The interval \((-r, r)\) represents the domain within which the power series is valid.

Ratio Test

The ratio test is a handy tool to determine the convergence or divergence of infinite series. Given a series \[ \sum a_k \] we examine the limit:\[ \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| = C \]If \(C < 1\), the series converges absolutely. If \(C > 1\), or the limit doesn't exist, the series diverges. If \(C = 1\), the ratio test is inconclusive.

When applying it to a power series, after calculating \(C\), we can derive conditions for convergence:

• If \(|x| < 1/C\), the series converges.
• If \(|x| = 1/C\), the test is inconclusive.
• If \(|x| > 1/C\), the series diverges.

This test helps find the radius of convergence, \(R\), by setting \(|x| < R\). For the series \(\sum a_k x^{2k}\), changing variables can alter the required threshold for \(x\), adjusting the radius to \(\sqrt{r}\).

Convergence of Series

Understanding the convergence of series is essential for analyzing many mathematical problems. A series like \[ \sum a_k \] converges if the sum approaches a particular finite value as the number of terms \(k\) approaches infinity. If it doesn't settle on a particular value, we say the series diverges.

For power series, not every choice of \(x\) leads to convergence. The series will only converge

• If \(|x|\) is within the radius of convergence. This set of values allows series to sum towards a stable number.
• If \(|x| > R\), the series diverges. Beyond this radius, the terms grow too large to sum up neatly.

Finding whether a series converges in its entirety or not often involves tests like the ratio test to set the bounds of \(x\) accordingly. Each series has its unique convergence function, influenced by its coefficients and terms.