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Magazine Chapter 9: Problem 31
Find the interval of convergence. $$\sum_{n=1}^{\infty} \frac{x^{2 n+1}}{n !}$$
Short Answer
Expert verified
The interval of convergence is \((-\infty, \infty)\).
Step by step solution
01
Understanding the Series
Identify the structure of the given series, which is \( \sum_{n=1}^{\infty} \frac{x^{2n+1}}{n!} \). This is similar to a power series except it has odd-powered terms (\( x^{2n+1} \)) and includes factorial in the denominator.
02
Applying the Ratio Test
To find the interval of convergence, apply the ratio test. Consider the terms \( a_n = \frac{x^{2n+1}}{n!} \) and \( a_{n+1} = \frac{x^{2(n+1)+1}}{(n+1)!} \). Calculate the limit of the ratio \( \left| \frac{a_{n+1}}{a_n} \right| \).
03
Calculating the Ratio Limit
The ratio \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{x^{2n+3}}{(n+1)!} \cdot \frac{n!}{x^{2n+1}} \right| = \left| \frac{x^2}{n+1} \right| \). Evaluate the limit as \( n \to \infty \): \( \lim_{n \to \infty} \left| \frac{x^2}{n+1} \right| = 0 \).
04
Determine Convergence
Since the limit is 0 for any value of \( x \), the series converges for all \( x \). Thus, the interval of convergence is \( (-\infty, \infty) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ratio Test
The Ratio Test is a powerful tool used to determine whether a series converges. It helps us understand if the infinite sum of terms in a series settles on a particular value. Apply the test by comparing consecutive terms in a sequence. To use the Ratio Test, compute the limit:
- Find the absolute value of the ratio of consecutive terms: \( \left| \frac{a_{n+1}}{a_n} \right| \).
- Evaluate the limit as the sequence progresses towards infinity: \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).
Power Series
A power series is an expression of the form \( \sum_{n=0}^{\infty} c_n (x - a)^n \). It resembles a polynomial, but unlike polynomials, it includes infinitely many terms. Each term is a power of \( x \) raised to a consecutive integer exponent. The major aspect of power series is that:
- They depend on the variable \( x \).
- They are defined in terms of powers, like \( x^n \).
- Like usual polynomials, they have coefficients \( c_n \).
- They may converge only over a specific range or interval of \( x \).
Factorial
The factorial, denoted as \( n! \), represents the product of all whole numbers from 1 to \( n \). For example, \( 5! \) equals 5 times 4 times 3 times 2 times 1, resulting in 120. Factorials are important in many mathematical concepts, especially in series and permutations, because:
- They help count arrangements or orderings of items.
- They show up in the denominator when calculating probability and combinations.
- In power series, factorials often appear in the denominator, affecting how fast the sequence terms decrease as \( n \) increases.
Convergence
Convergence in the context of series involves determining whether a series sums to a finite value. When contemplating convergence, consider two things:
- If taking the sum of its terms brings it close to a specific number, known as the limit, the series converges.
- If it doesn't settle on a particular number and keeps growing indefinitely, it diverges.
Series
A series is essentially the sum of the terms of a sequence. Imagine listing numbers in order, then summing them up consecutively:
- If we list terms of finite sequences, we get finite series.
- For infinite series, we keep on adding infinitely many terms.
- Series are used to approximate complex functions, such as exponential functions or trigonometric functions.
- Understanding series helps in grasping advanced calculus concepts efficiently.
- They bring context to complex systems in engineering, physics, and economics.
