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Chapter 6: Problem 21

Find the radius of convergence \(R\) and interval of convergence for \(\sum a_{n} x^{n}\) with the given coefficients \(a_{n}\). $$ \sum_{n=1}^{\infty} \frac{10^{n} x^{n}}{n !} $$

Short Answer

Expert verified
The radius of convergence is \( R = \infty \) and the interval is \(( -\infty, \infty )\).

Step by step solution

01

Identify the series

The series given is \( \sum_{n=1}^{\infty} \frac{10^{n} x^{n}}{n!} \). We notice that it is of the form \( \sum a_n x^n \) with coefficients \( a_n = \frac{10^n}{n!} \).
02

Apply the Ratio Test

To find the radius of convergence \( R \), we use the Ratio Test. This involves finding the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). The series converges when \( L < 1 \).
03

Calculate the ratio \( \frac{a_{n+1}}{a_n} \)

With \( a_n = \frac{10^n}{n!} \), calculate \( a_{n+1} = \frac{10^{n+1}}{(n+1)!} \). Thus, \( \frac{a_{n+1}}{a_n} = \frac{10^{n+1} \cdot n!}{10^n \cdot (n+1)!} = \frac{10}{n+1} \).
04

Compute the limit \( L \)

Compute the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{10}{n+1} = 0 \). Since this limit is 0, which is less than 1, the series converges for all \( x \).
05

Determine the radius of convergence \( R \)

Since \( L = 0 < 1 \) for any \( x \), the radius of convergence \( R \) is \( \infty \). This means the series converges for all \( x \).
06

Establish the interval of convergence

The interval of convergence, given the radius \( R = \infty \), is \(( -\infty, \infty )\). The series converges for any real number \( x \).

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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 key tool for determining the convergence of series, especially for power series. Its purpose is to evaluate whether a series converges absolutely. To apply the Ratio Test, we need to find the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). If this limit \( L < 1 \), the series converges absolutely. If \( L > 1 \) or if \( L \) is infinite, the series diverges. When \( L = 1 \), the test is inconclusive.
  • Calculate \( \frac{a_{n+1}}{a_n} \).
  • Determine the limit as \( n \to \infty \).
  • Use the result to find out about convergence.
This test is particularly effective in series with factorials or exponential terms, which simplify nicely when taking ratios.
Power Series
A power series is a type of infinite series characterized by the expression \( \sum a_n x^n \). It has coefficients \( a_n \) and is composed of variable terms \( x^n \), which means it can be analyzed to find where it converges. These series provide ways to approximate functions and solve problems in calculus and analysis. In a power series, each term involves powers of a variable multiplied by coefficients.
  • Looks like \( a_0 + a_1x + a_2x^2 + \ldots \)
  • Useful in approximating functions.
  • Convergence depends on the values of \( x \).
Understanding how a power series converges requires knowing the radius and interval of convergence.
Interval of Convergence
The interval of convergence defines the set of \( x \) values for which a power series converges. To establish this interval, we'll typically first find the radius of convergence using tests like the Ratio Test. Once the radius \( R \) is known, the interval can be determined as the range \( (c-R, c+R) \) where \( c \) is the center of the series, often \( 0 \) in standard series.
  • Calculate the radius \( R \).
  • Center the interval around the series center.
  • Test endpoint convergence if necessary.
In some cases, like the given series where \( R = \infty \), the interval is the entire set of real numbers, \( (-\infty, \infty) \). This implies the series converges for all real \( x \).
Convergence of Series
Understanding the convergence of a series is crucial for determining its behavior and applicability. Convergence implies that as we sum an infinite number of terms, the total approaches a specific value. Various tests, including the Ratio Test, help us ascertain where convergence occurs.
  • The series converges if the sum approaches a limit.
  • The concept of absolute convergence ensures every rearrangement of the series also converges.
  • In regular tests, studying endpoints is important when the interval of convergence is finite.
Every power series presents a potential range (or interval) in which its terms consolidate toward a finite sum based on specific values of \( x \). In the example series, absolute convergence everywhere simplifies the analysis, leading to applications over all real numbers.

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Most popular questions from this chapter

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