Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-8\), find the convergence set for the given power series. $$ \sum_{n=1}^{\infty} \frac{(x+1)^{n}}{n !} $$

Step-by-Step Solution

Answer

The series converges for all \( x \).

Step 1: Understand the Power Series

The given power series is composed as \( \sum_{n=1}^{\infty} \frac{(x+1)^{n}}{n!} \). This follows the general form of a power series \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( a_n = \frac{1}{n!} \) and \( c = -1 \), which is our center.

Step 2: Use the Ratio Test for Convergence

Apply the ratio test to determine the interval of convergence. The ratio test states that for a series \( \sum a_n \), if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \), then the series converges if \( L < 1 \), diverges if \( L > 1 \), and is inconclusive if \( L = 1 \). In this case, we need to find \( \lim_{n \to \infty} \left| \frac{(x+1)^{n+1}}{(n+1)!} \cdot \frac{n!}{(x+1)^n} \right| \).

Step 3: Simplify the Ratio

Plug in the terms from the series into the ratio from the Ratio Test: \[ \lim_{n \to \infty} \left| \frac{(x+1)^{n+1}}{(n+1)!} \cdot \frac{n!}{(x+1)^n} \right| = \lim_{n \to \infty} \left| \frac{(x+1) \cdot (x+1)^n}{(x+1)^n} \cdot \frac{1}{n+1} \right| \]. Simplify to:\[ \lim_{n \to \infty} \left| \frac{x+1}{n+1} \right| \].

Step 4: Evaluate the Limit for Convergence

Evaluate the limit:\[ \lim_{n \to \infty} \left| \frac{x+1}{n+1} \right| = 0 \]. As \( n \to \infty \), the division by a growing number \( n+1 \) ensures the expression approaches 0. Since 0 is less than 1, the Ratio Test confirms the series converges for all \( x \).

Step 5: Conclusion on Convergence Set

Since the limit evaluates to 0 regardless of the value of \( x \), the series converges for any real number \( x \). This means the interval of convergence is all real numbers: \( (-\infty, \infty) \).

Key Concepts

Ratio Test

The Ratio Test is a popular method used to determine the convergence of an infinite series. It's especially useful for dealing with series that include factorials or exponentials. In general, the ratio test takes the form of evaluating the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L \).

• If \( L < 1 \), the series converges absolutely.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive and other methods must be used.

Consider the power series from the exercise: \( \sum_{n=1}^{\infty} \frac{(x+1)^{n}}{n!} \). Here, the Ratio Test simplifies to considering the expression \( \lim_{n \to \infty} \left| \frac{x+1}{n+1} \right| \). As you plug in values and simplify, the special presence of \( n! \) and its continual growth ensures that the limit reduces to zero. Since 0 < 1, this guarantees convergence for any value of \( x \). The Ratio Test is powerful here because it quickly shows that any real number will work as \( x \).

Interval of Convergence

The Interval of Convergence represents all the values of \( x \) for which a series converges. For a power series, it is crucial to determine this because it tells us where the series behaves nicely, providing meaningful sums. The exercise in focus uses the Ratio Test to find the interval of convergence. Upon applying the test, the limit was \( 0 \) irrespective of \( x \), indicating convergence everywhere on the real number line. Thus, the series converges for any possible \( x \), leading to an Interval of Convergence of \( (-\infty, \infty) \).

• This means the series behaves nicely and sums to a finite value regardless of how large or small \( x \) is.

Such convergence without restriction is typical for series similar to the exponential series \( e^x \), which happens here since factorial growth in the denominator contributes significantly to convergence.

Power Series Analysis

Power Series Analysis involves examining and understanding the behavior of series expressed in the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \). This kind of series is central in calculus and mathematical analysis because it approximates functions and can solve differential equations. In the given problem, we analyze the power series \( \sum_{n=1}^{\infty} \frac{(x+1)^{n}}{n!} \). Recognizing the general form helps in identifying conversion techniques, particularly using derivative and integral operations applied to series terms.

• Here, the "center" of the series is \( c = -1 \), meaning it's shifted horizontally along the \( x \)-axis.
• Each term \( \frac{(x+1)^{n}}{n!} \) shares structural similarity with exponential functions, revealing much about the nature of the series.

Through analysis, one confirms the series converges everywhere, drawing parallels to entire functions like \( e^{x+c} \). This makes power series a robust tool in mathematical modeling and theoretical comprehension.