Ratio Test Power Series

The Ratio Test for Power Series: Determining Convergence and Radius of Convergence

Power series, infinite sums of the form $\sum_{n=0}^{\infty} c_n(x-a)^n$, are fundamental objects in calculus and analysis. Understanding their convergence is crucial for many applications. While various tests exist, the ratio test provides a particularly elegant and powerful method for determining the interval of convergence of a power series, specifically its radius of convergence. This article will explore the ratio test's application to power series, explaining its mechanics and illustrating its use through examples.

Understanding the Ratio Test

The ratio test examines the limit of the ratio of consecutive terms in a series. For a general series $\sum_{n=0}^{\infty} a_n$, the ratio test states:

1. If $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = L < 1$, the series converges absolutely.
2. If $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = L > 1$ or $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = \infty$, the series diverges.
3. If $\lim_{n\to\infty} \left|\frac{a_{n+1}}{a_n}\right| = L = 1$, the test is inconclusive.

Applying the Ratio Test to Power Series

When applying the ratio test to a power series $\sum_{n=0}^{\infty} c_n(x-a)^n$, we treat the terms $a_n = c_n(x-a)^n$. The ratio becomes:

$\left|\frac{a_{n+1}}{a_n}\right| = \left|\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}\right| = |x-a| \left|\frac{c_{n+1}}{c_n}\right|$

The limit as $n \to \infty$ then depends on the behavior of $\left|\frac{c_{n+1}}{c_n}\right|$. Let's denote:

$R = \lim_{n\to\infty} \left|\frac{c_n}{c_{n+1}}\right|$ (Note: this is the reciprocal of the usual limit). This limit, R, represents the radius of convergence.

Determining the Radius and Interval of Convergence

Using the ratio test on the power series, we find that the series converges absolutely when:

$|x-a| \lim_{n\to\infty} \left|\frac{c_{n+1}}{c_n}\right| < 1$

This simplifies to:

$|x-a| < R$

This inequality defines an interval centered at a with a radius of R. The interval of convergence is then (a - R, a + R). We must also test the endpoints, x = a - R and x = a + R, separately using other convergence tests (e.g., the alternating series test, p-series test) since the ratio test is inconclusive at these points.

Example: Finding the Radius and Interval of Convergence

Let's consider the power series: $\sum_{n=1}^{\infty} \frac{x^n}{n^2}$

Here, $c_n = \frac{1}{n^2}$, $a = 0$. We compute:

$\lim_{n\to\infty} \left|\frac{c_n}{c_{n+1}}\right| = \lim_{n\to\infty} \left|\frac{\frac{1}{n^2}}{\frac{1}{(n+1)^2}}\right| = \lim_{n\to\infty} \left(\frac{n+1}{n}\right)^2 = 1$

Therefore, R = 1. The interval of convergence is (-1, 1). Now we test the endpoints:

x = -1: $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$ converges absolutely (by the alternating series test).
x = 1: $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges (p-series with p = 2 > 1).

Thus, the interval of convergence is [-1, 1].

Limitations of the Ratio Test

The ratio test is a powerful tool, but it has limitations. As mentioned earlier, if the limit of the ratio is 1, the test is inconclusive. In such cases, other convergence tests are needed. Furthermore, the ratio test can be computationally challenging for power series with complex coefficients or intricate patterns in their terms.

Summary

The ratio test provides an efficient method for determining the radius and interval of convergence of a power series. By examining the limit of the ratio of consecutive terms, we can identify the radius of convergence, R. The interval of convergence is then (a - R, a + R), with the endpoints needing separate analysis using other convergence tests. While powerful, the ratio test is not universally applicable, and its limitations must be considered.

FAQs

1. What if the limit of the ratio is 1? If the limit is 1, the ratio test is inconclusive. Other convergence tests, such as the root test, comparison test, or integral test, must be employed.

2. Can the radius of convergence be infinite? Yes, if the limit of the ratio is 0, the radius of convergence is infinite, meaning the power series converges for all real numbers.

3. What does the radius of convergence represent geometrically? The radius of convergence represents the radius of the largest open interval centered at 'a' for which the power series converges absolutely.

4. Why do we need to test the endpoints separately? The ratio test is inconclusive at the endpoints of the interval of convergence. The series might converge conditionally or diverge at these points. Other tests are necessary to determine the convergence at the endpoints.

5. What are some alternative tests for convergence besides the ratio test? The root test, comparison test, limit comparison test, integral test, and alternating series test are some alternatives useful for determining convergence. The choice of test depends on the specific series being examined.

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Power series (Sect. 10.7) Power series deﬁnition and exampl power series centered at x0 = 0 is y (x) = . + c1 x + c2 x2 + c3. |x| < 1. = 1 + (x − 1) + + · · · . n! 2! n=0 ∞ (−1) = , that is, y (x) = x(2n+1), (2n + 1)! . tion: The power series y (x) is a geometric series for x ∈ R. Geometric series converge for |x| < 1, and. am. t. n . x. x ∈ R . the. n→∞ a. and for x ∈ (−∞, −1) ∪ (1, �.

Differential Equations - Review : Power Series - Pauls Online … 16 Nov 2022 · In this section we give a brief review of some of the basics of power series. Included are discussions of using the Ratio Test to determine if a power series will converge, adding/subtracting power series, differentiating power series and index shifts for power series.

Power Series - UC Davis By the ratio test, the power series converges if 0 ≤ r<1, or |x− c| <R, and diverges if 1 <r≤ ∞, or |x−c| >R, which proves the result. The root test gives an expression for the radius of convergence of a general power series. Theorem 6.5 (Hadamard). The radius of convergence Rof the power series ∑∞ n=0 an(x−c)n is given by R= 1 ...

THE RATIO TEST - Reed College THE RATIO TEST Consider a complex power series all of whose coe cients are nonzero, f(z) = X1 n=0 a n(z c)n; a n 6= 0 for each n: Suppose that the limit R = R(f) = lim n!1 ja nj ja n+1j exists in the extended nonnegative real number system [0;1]. We show that R is the radius of convergence of f, f(z) converges absolutely on the open disk of ...

Ratio Test – Definition, Conditions, and Examples on Series It’s one of the first tests used when assessing the convergence or divergence of a given series – especially the Taylor series. The ratio test can also help us in finding the interval and radius of the interval of a power series making it a very important convergence test.

9.6: Ratio and Root Tests - Mathematics LibreTexts 18 Oct 2018 · Use the ratio test to determine absolute convergence of a series. Use the root test to determine absolute convergence of a series. Describe a strategy for testing the convergence of a given series. In this section, we prove the last two series convergence tests: the …

Calculus II - Power Series - Pauls Online Math Notes 16 Nov 2022 · In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series. We will also illustrate how the Ratio Test and Root Test can be used to determine the radius and interval of convergence for a power series.

(PS1 and PS2) Power Series — Calculus 2 - blue tangent The Ratio Test is applicable for all $x$-values except the center value of our power series. For this problem, the center value is $a=3$ . This means that when we calculated the above limit, it was actually for $x\neq 3$ which is equivalent to $|x-3|\neq 0$ .

Ratio test - Wikipedia In mathematics, the ratio test is a test (or "criterion") for the convergence of a series ∑ n = 1 ∞ a n , {\displaystyle \sum _{n=1}^{\infty }a_{n},} where each term is a real or complex number and a n is nonzero when n is large.