Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine convergence or divergence for each of the series. Indicate the test you use. $$ \sum_{n=1}^{\infty} \frac{n^{n}}{(2 n) !} $$

Step-by-Step Solution

Answer

The series converges by the ratio test.

Step 1: Identify the series

The given series is \( \sum_{n=1}^{\infty} \frac{n^{n}}{(2 n)!} \). We need to determine whether this series converges or diverges.

Step 2: Consider a suitable test

For a series of the form \( \frac{an}{bn} \), where the factorials are involved, the ratio test is often suitable. The ratio test can help determine if a series converges.

Step 3: Apply the Ratio Test

The ratio test states that a series \( \sum a_n \) converges if \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \). For our series, let \( a_n = \frac{n^n}{(2n)!} \). We find \( a_{n+1} = \frac{(n+1)^{n+1}}{(2(n+1))!} \).

Step 4: Compute \( \frac{a_{n+1}}{a_n} \)

Compute:\[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^{n+1}}{(2n+2)!} \times \frac{(2n)!}{n^n} = \frac{(n+1)^{n+1} \times (2n)!}{n^n \times (2n+2)!} \]

Step 5: Simplify the expression

Simplify the factorial term:\[ \frac{(2n)!}{(2n+2)!} = \frac{1}{(2n+1)(2n+2)} \]Thus, \[ \frac{a_{n+1}}{a_n} = \frac{(n+1)^{n+1}}{n^n} \times \frac{1}{(2n+1)(2n+2)} \]

Step 6: Evaluate the limit

Evaluate the limit as \( n \to \infty \):\[ \lim_{n \to \infty} \left| \frac{n+1}{n} \right|^n \times \frac{n+1}{(2n+1)(2n+2)} \]The first term \( \left( 1 + \frac{1}{n} \right)^n \) approaches \( e \), so:\[ \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 0 \] because the factorial grows much faster than any polynomial.

Step 7: Conclude the test

Since \( \lim_{n \to \infty} \frac{a_{n+1}}{a_n} = 0 < 1 \), by the ratio test, the series converges.

Key Concepts

Ratio Test

The Ratio Test is a valuable tool in determining the convergence or divergence of an infinite series. This test is particularly handy when working with series involving factorials or exponential terms. The idea is to compare the ratio of successive terms in the series.

For a given series \( \sum a_n \), the ratio test investigates the limit \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If this limit is less than 1, the series converges absolutely. If the limit is greater than 1, the series diverges. If it equals 1, the test is inconclusive. This test simplifies complex series evaluation by focusing on the growth rate of terms in the series relative to each other.

• Apply the test to series with factorial terms, like \( \frac{n^{n}}{(2n)!} \).
• Simplify the ratio of terms to identify convergence or divergence.
• Use the limit result \( < 1 \) to confirm convergence.

By leveraging the Ratio Test, one can efficiently assess the behavior of challenging infinite series.

Factorial Function

The factorial function is a crucial element in many mathematical situations and series tests. Defined as \( n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 \), factorials grow extremely fast, much faster than any polynomial function. This rapid growth is key when analyzing series like \( \sum \frac{n^n}{(2n)!} \).

In the context of convergence, factorial growth often overshoots polynomial and exponential growth over the long term. For instance, the term \( (2n)! \) in our example series grows much more rapidly than \( n^n \), leading to a diminishing overall series term as \( n \to \infty \).

• Factorials denote the product of an integer sequence.
• They exhibit exponential growth behavior in series.
• Crucial for applying Ratio Test to judge series convergence.

Understanding the nature of factorial functions is essential to employing convergence tests effectively.

Infinite Series

Infinite series are expressions that sum an infinite sequence of terms. They play a vital role in various fields of mathematics and are essential in calculus and analysis. One must often determine if a series converges (approaches a finite sum) or diverges (grows without bound).

Analyzing series like \( \sum_{n=1}^{\infty} \frac{n^{n}}{(2n)!} \) involves using convergence tests, including the Ratio Test. The behavior of an infinite series depends on the characteristics of its terms, such as polynomial or factorial functions.

• The goal is to determine if a series has a finite sum.
• Convergence tests help analyze series behavior.
• Important in modeling and solving real-world problems.

Infinite series allow us to approximate functions and describe complex phenomena, making their study a cornerstone of advanced mathematics.