Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ \frac{x+5}{1 \cdot 2}+\frac{(x+5)^{2}}{2 \cdot 3}+\frac{(x+5)^{3}}{3 \cdot 4}+\frac{(x+5)^{4}}{4 \cdot 5}+\cdots $$

Step-by-Step Solution

Answer

The convergence set for the series is \(-6 < x < -4\).

Step 1: Identify the General Term

To find the general term, we observe the given series and notice that the nth term has the form \( \frac{(x+5)^n}{n(n+1)} \). Thus, the general term for the power series is \( a_n = \frac{(x+5)^n}{n(n+1)} \).

Step 2: Apply the Ratio Test

To find the convergence set, we apply the Ratio Test. For the series \( \sum a_n \), the Ratio Test states we must examine the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Step 3: Calculate the Ratio Expression

Evaluate \( \frac{a_{n+1}}{a_n} = \frac{(x+5)^{n+1}}{(n+1)(n+2)} \cdot \frac{n(n+1)}{(x+5)^n} = \frac{(x+5)\cdot n}{(n+2)} \).

Step 4: Simplify the Ratio

Simplify further to get \( \frac{a_{n+1}}{a_n} = \frac{(x+5)\cdot n}{(n+2)} \). As \( n \to \infty \), this simplifies to \( |x+5| \), so \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = |x+5| \).

Step 5: Determine Convergence

For the series to converge using the Ratio Test, the condition \( |x+5| < 1 \) must be satisfied. Thus, the series converges when \( -1 < x+5 < 1 \).

Step 6: Find the Convergence Interval

Solving \( -1 < x+5 < 1 \), we subtract 5 throughout: \( -6 < x < -4 \). This is the convergence interval.

Key Concepts

Ratio Test

The Ratio Test is an essential tool in determining the convergence of an infinite series. It evaluates the limit of the absolute value of the ratio of consecutive terms as the number of terms goes to infinity. By this test, for a series \( \sum a_n \), you consider the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
The rules for convergence are straightforward:

• If the limit is less than 1, the series absolutely converges.
• If the limit is greater than 1, the series diverges.
• If the limit equals 1, the test is inconclusive.

To apply it, identify the general term of your series, create the ratio of \( a_{n+1} \) to \( a_n \), and simplify. This method is robust for series with factorial expressions or exponential growth, making it a widely-used approach in calculus.

General Term of a Series

Finding the general term of a series \( a_n \) is crucial for analyzing its properties, such as convergence. The general term expresses the nth element in terms of \( n \) and any variable present, like \( x \) in a power series.
In our series, after examining the pattern, the general term was identified as \( a_n = \frac{(x+5)^n}{n(n+1)} \).
This involves recognizing patterns in the numerators and denominators of each term:

• The numerator follows a pattern using powers of a transformed variable \((x+5)\).
• The denominator in this scenario evolves as \( n(n+1) \), hinting at a product of consecutive integers.

The process of identifying this term is foundational, enabling further tests like the Ratio Test to determine convergence.

Convergence Interval

A convergence interval is a range of values for which a power series converges. This range is determined after applying the Ratio Test and calculating the limit condition for convergence.
For a series \( |x+5| < 1 \), this expresses the condition \(-1 < x+5 < 1\). Solving for \( x \) gives us the convergence interval. In our example, we subtract 5 from all parts of the inequality, resulting in \(-6 < x < -4\).
The convergence interval indicates where the series behaves nicely and sums up to a finite limit. Always check end points separately when computing sums to know if the convergence holds there or not.

Absolute Convergence

Absolute convergence is a strong form of convergence wherein a series converges by considering the absolute values of its terms. This is important because absolute convergence implies regular convergence, though the converse might not hold.
To check for absolute convergence using the Ratio Test or other techniques, take the absolute values of the terms and determine if the resulting series converges.

• If a series \( \sum |a_n| \) converges, the original series \( \sum a_n \) converges absolutely.
• If absolute convergence is shown, rearranging terms doesn’t affect the sum, unlike conditional convergence.

Understanding absolute convergence helps in cases where the convergence type influences operations on the series or further analysis.