Find The Interval Of Convergence For The Given Power Series

Finding the Interval of Convergence for Power Series: A Comprehensive Guide

Determining the interval of convergence for a power series is a crucial step in understanding its behavior and applications in calculus and analysis. A power series, centered at a point a, is an infinite series of the form:

∑_n=0^∞ c_n(x - a)ⁿ = c₀ + c₁(x - a) + c₂(x - a)² + c₃(x - a)³ + ...

where c_n are constants and x is a variable. The interval of convergence is the set of all x-values for which the series converges. This interval can be an open interval, a closed interval, a half-open interval, or even a single point. Understanding how to find this interval is essential for various mathematical applications.

Understanding Convergence Tests

Before diving into the process of finding the interval of convergence, let's review some essential convergence tests. These tests help determine whether an infinite series converges or diverges. The most commonly used tests for power series include:

1. The Ratio Test

The ratio test is a powerful tool for determining the convergence of a series. For a series ∑a_n, the ratio test states:

• If lim_n→∞ |a_n+1/a_n| < 1, the series converges absolutely.
• If lim_n→∞ |a_n+1/a_n| > 1, the series diverges.
• If lim_n→∞ |a_n+1/a_n| = 1, the test is inconclusive.

For power series, we apply the ratio test to the terms c_n(x - a)ⁿ.

2. The Root Test

Similar to the ratio test, the root test examines the limit of the nth root of the absolute value of the terms. For a series ∑a_n:

• If lim_n→∞ |a_n|^1/n < 1, the series converges absolutely.
• If lim_n→∞ |a_n|^1/n > 1, the series diverges.
• If lim_n→∞ |a_n|^1/n = 1, the test is inconclusive.

The root test can be particularly useful when dealing with series containing terms raised to the power of n.

3. The Comparison Test

The comparison test compares the terms of a given series to the terms of a known convergent or divergent series. If 0 ≤ a_n ≤ b_n for all n and ∑b_n converges, then ∑a_n also converges. Conversely, if 0 ≤ b_n ≤ a_n for all n and ∑b_n diverges, then ∑a_n also diverges.

4. The Limit Comparison Test

This test is a refinement of the comparison test and is useful when a direct comparison is difficult. If lim_n→∞ a_n/b_n = L, where L is a finite positive number, then ∑a_n and ∑b_n either both converge or both diverge.

Step-by-Step Procedure for Finding the Interval of Convergence

Let's outline a systematic approach for finding the interval of convergence for a power series:

1. Apply a Convergence Test: Typically, the ratio test is the most convenient method for power series. Apply the ratio test to the series ∑c_n(x - a)ⁿ. Calculate the limit:

   lim_n→∞ |[c_n+1(x - a)ⁿ⁺¹] / [c_n(x - a)ⁿ]| = lim_n→∞ |(c_n+1/c_n)(x - a)|

2. Determine the Radius of Convergence: The limit from step 1 will typically result in an expression involving |x - a|. For the series to converge, this limit must be less than 1:

   lim_n→∞ |(c_n+1/c_n)(x - a)| < 1

   Solve this inequality for |x - a|. This will give you the radius of convergence, R. The radius of convergence represents the distance from the center a to the endpoints of the interval of convergence. The interval of convergence is centered at a, with a radius of R.

3. Check the Endpoints: Once you've found the radius of convergence, you must check the convergence of the series at the endpoints of the interval. Substitute the endpoints into the original power series and test for convergence using other convergence tests like the p-series test, alternating series test, or integral test, as appropriate.

4. State the Interval of Convergence: Based on the convergence at the endpoints, you can state the interval of convergence. The interval can be open (e.g., (-R, R)), closed (e.g., [-R, R]), or half-open (e.g., (-R, R] or [-R, R)).

Examples

Let's illustrate this process with a few examples:

Example 1: Find the interval of convergence for the power series ∑_n=1^∞ (xⁿ/n).

1. Ratio Test:

   lim_n→∞ |[(xⁿ⁺¹)/(n+1)] / [(xⁿ)/n]| = lim_n→∞ |x(n/(n+1))| = |x|

2. Radius of Convergence: For convergence, |x| < 1, so the radius of convergence R = 1.

3. Check Endpoints:

   • x = 1: The series becomes ∑_n=1^∞ (1/n), which is the harmonic series and diverges.
   • x = -1: The series becomes ∑_n=1^∞ (-1)ⁿ/n, which converges by the alternating series test.

4. Interval of Convergence: The interval of convergence is [-1, 1).

Example 2: Find the interval of convergence for the power series ∑_n=0^∞ (x - 2)ⁿ / (n!)

1. Ratio Test:

   lim_n→∞ |[(x-2)ⁿ⁺¹/(n+1)!] / [(x-2)ⁿ/n!]| = lim_n→∞ |(x-2)/(n+1)| = 0

Since the limit is 0 for all x, the series converges for all x.

2. Radius and Interval of Convergence: The radius of convergence is infinite (R = ∞), and the interval of convergence is (-∞, ∞).

Example 3: Find the interval of convergence for the power series ∑_n=0^∞ (n!(x+1)ⁿ)

1. Ratio Test:

   lim_n→∞ |[(n+1)!(x+1)ⁿ⁺¹]/[n!(x+1)ⁿ]| = lim_n→∞ |(n+1)(x+1)|

This limit is only less than 1 if x+1 = 0, meaning x = -1.

2. Radius and Interval of Convergence: The radius of convergence is 0 (R = 0), and the interval of convergence is just the single point x = -1. The series only converges at x = -1.

These examples demonstrate the application of the ratio test, endpoint analysis, and the resulting intervals of convergence. Remember to always rigorously check the endpoints, as convergence behavior at these points can significantly affect the final interval. Choosing the appropriate convergence test is crucial for efficiency and accuracy. The choice often depends on the specific form of the series terms. Understanding these techniques is fundamental to mastering power series analysis.