Question

In Exercises \(7-28,\) find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{(n+1)(n+2)} $$

Short answer

The interval of convergence of the power series \(\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{(n+1)(n+2)}\) is \([-1, 1]\).

Step-by-step solution

Apply the Ratio Test for absolute convergence

First, we want to find the absolute convergence of the series by applying the ratio test. The Ratio Test states that if \(\lim_{{n \to \infty}} \left| \frac{{a_{n+1}}}{{a_n}} \right| = L\), then if \( L < 1\), the series converges absolutely, if \( L > 1 \) the series diverges, and if \( L = 1 \) the test is inconclusive. In this case, our \( a_n \) is \(\frac{(-1)^{n} x^{n}}{(n+1)(n+2)}\), which will give \(\lim_{{n \to \infty}} \left| \frac{{(-1)^{n+1} x^{n+1}}{(n+2)(n+3)}}{\frac{{(-1)^n x^n}}{(n+1)(n+2)}} \right| = \lim_{{n \to \infty}} \left| \frac{{(-1) x}}{(n+3)} \right| = |x| \lim_{{n \to \infty}} \frac{1}{{n+3}} = 0 \) for -1 < x < 1.

Determine the endpoints for conditional convergence

Next, we need to check the endpoints \(x = -1\) and \(x = 1\) separately. We substitute \(x = -1\) and \(x = 1\) into the series and check for convergence. Let's start with \(x = -1\). Substitute \(x = -1\) into the series , then we get \(\sum_{n=0}^{\infty} \frac{1}{(n+1)(n+2)}\), which is a convergent p-series. Now we substitute \(x = 1\) into the series and get \(\sum_{n=0}^{\infty} \frac{{(-1)^n}}{(n+1)(n+2)}\), which is an alternating series. The absolute value of the terms is decreasing and approaching zero, so the series converges by the Alternating Series Test.

State the Interval of Convergence

After finding the absolute convergence from ratio test and conditional convergence at endpoints, we state the interval of convergence. The power series \(\sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{(n+1)(n+2)}\) has an interval of convergence of \([-1, 1]\).

Key concepts

Power Series

A power series is an infinite series of the form \(\sum_{n=0}^\infty a_n(x-c)^n\), where \(a_n\) represents the coefficient of the nth term, \(x\) is the variable, and \(c\) is the center of the series. These series can be incredibly useful for representing functions as they can converge within a certain radius around point \(c\), known as the interval of convergence. In simple terms, this interval tells us the range of \(x\) values for which the series will accurately represent a function.

In the given exercise, the power series is centered around \(c = 0\), and it becomes essential to find out where this series converges, meaning the power series behaves nicely, summing up to a finite value. When checking convergence, we typically exclude the center \(c\), and instead focus on the stretch of \(x\) values moving away from \(c\), which in this exercise ranges from -1 to 1 according to the Ratio Test.

Ratio Test

The Ratio Test is a criterion used to predict the convergence or divergence of infinite series, especially useful for power series and series with factorials, exponentials, or nth powers. To apply the Ratio Test, you take the limit of the absolute value of the ratio of successive terms, denoted as \(\lim_{n \to \infty} \left| \frac{{a_{n+1}}}{{a_n}} \right|\).

Here's how the Ratio Test works:

• If the limit \(L < 1\), the series converges absolutely.
• If the limit \(L > 1\), the series diverges.
• If the limit \(L = 1\), the test is inconclusive, meaning it doesn't tell us whether the series converges or diverges.

The Ratio Test provides the key to unlocking the interval of convergence for a power series. In the exercise, the Ratio Test determined that the series converges when \(|x| < 1\), a crucial step in finding the interval of convergence.

Alternating Series Test

The Alternating Series Test is another method for determining the convergence of series, specifically dealing with series whose terms alternate in sign. An alternating series is of the form \(-1^n a_n\), where the term \(a_n\) is non-negative. This test states that if the absolute values of the terms decrease monotonically (each term is less than the one before it) and the limit of the sequence of terms is zero, then the alternating series converges.

Applying the Alternating Series Test involves checking two conditions:

• The terms \(a_n\) must decrease with each successive n.
• The limit of the terms \(a_n\) as n approaches infinity must be zero.

If both conditions are met, you can confidently say the series converges. This test was used at the endpoints in the exercise, where \(x = -1\) created a p-series and \(x = 1\) an alternating series. After evaluation, both conditions of the Alternating Series Test were satisfied for \(x = 1\), confirming convergence at this endpoint.