Problem 11 - Thomas Calculus

Question

In Exercises \(1-36,\) (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely (c) conditionally? $$ \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{n}}{n !} $$

Step-by-Step Solution

Answer

The series converges absolutely for all \(x\); there is no conditional convergence.

Step 1: Identify the series

The given series is \(\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n!}\). This is a power series centered at 0 with coefficients \(a_n = \frac{(-1)^n}{n!}\).

Step 2: Use the Ratio Test for Convergence

To find the radius of convergence, apply the Ratio Test. Compute the limit \( L \) of the absolute value of the ratio of successive terms:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1} x^{n+1}}{a_n x^n} \right| = \lim_{n \to \infty} \left| \frac{(-1)^{n+1} x^{n+1}}{(n+1)!} \times \frac{n!}{(-1)^n x^n} \right| = \lim_{n \to \infty} \left| \frac{x}{n+1} \right| \]As \(n \to \infty\), \(\frac{|x|}{n+1} \to 0\) for any \(x\). Thus, \(L = 0\), and the power series converges for all \(x\).

Step 3: Determine the Radius and Interval of Convergence

Since \(L = 0\) is less than 1 for any \(x\), the series converges for all real numbers \(x\). Therefore, the radius of convergence \(R = \infty\), and the interval of convergence is \((-\infty, \infty)\).

Step 4: Check for Absolute Convergence

The series is absolutely convergent if \(\sum_{n=0}^{\infty} \left| \frac{(-1)^n x^n}{n!} \right| = \sum_{n=0}^{\infty} \frac{|x|^n}{n!}\) converges. Since this is the exponential series \(e^{|x|}\), which converges for all \(x\), the original series converges absolutely for all \(x\).

Step 5: Assess Conditional Convergence

Because the series converges absolutely for all \(x\), there are no \(x\) values for which the series converges merely conditionally. Conditional convergence occurs when a series converges but does not converge absolutely.

Key Concepts

Power Series

A power series is a type of infinite series that has the form:

• \[\sum_{n=0}^{\infty} a_n (x-c)^n\]

The series sums terms involving a variable raised to varying powers, multiplied by coefficients. Each term in the series takes the form of \(a_n (x-c)^n\), where \(a_n\) is a sequence of coefficients, \(x\) is the variable, and \(c\) is the center of the series. In the given series \(\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n!}\),

• \(a_n = \frac{(-1)^n}{n!}\)
• center \(c = 0\)

Power series are particularly interesting because they act like polynomials with, potentially, infinitely many terms. Power series can be used to represent and analyze functions over certain intervals, depending on their convergence behavior, which is determined by their radius of convergence.

Ratio Test

The Ratio Test is a method used to determine the convergence or divergence of an infinite series. Here's how it works:

• You have to calculate the limit:\[L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|\]where \(a_n\) are the terms of the series.
• Convergence criteria using the Ratio Test:
  • If \(L < 1\), the series converges absolutely.
  • If \(L > 1\) or \(L = \infty\), the series diverges.
  • If \(L = 1\), the test is inconclusive.

For the series \(\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n!}\), the Ratio Test gives a limit:\[L = \lim_{n \to \infty} \left| \frac{x}{n+1} \right| = 0\]Since this is less than 1, it means the series absolutely converges for all values of \(x\).

Absolute Convergence

A series \(\sum a_n\) is said to converge absolutely if the series of absolute values \(\sum |a_n|\) also converges. Absolute convergence is a stronger form of convergence and provides more robustness in mathematical analysis:

• If a series converges absolutely, it also converges.
• This allows us to rearrange the terms without affecting the sum of the series.

For the given series, we examined absolute convergence by assessing:\[\sum_{n=0}^{\infty} \left| \frac{(-1)^n x^n}{n!} \right| = \sum_{n=0}^{\infty} \frac{|x|^n}{n!}\]This is the well-known exponential series \(e^{|x|}\), which is convergent for all \(x\). Therefore, the series converges absolutely for all values of \(x\). Absolute convergence implies regular convergence, but not the other way around.

Conditional Convergence

Conditional convergence refers to a peculiar situation where a series converges, but it does not converge absolutely. This normally implies that removing the alternating signs would lead to a divergent series. Conditional convergence often involves manipulation of the terms potentially affecting outcomes. Examples include the famous

• Alternating Series Test

and the way alternating series can sometimes converge even when their corresponding non-alternating series diverge.In this case, however, the series \(\sum_{n=0}^{\infty} \frac{(-1)^n x^n}{n!}\) doesn't have any values of \(x\) for which it is only conditionally convergent because it converges absolutely for all \(x\), given that the series

• \(\sum_{n=0}^{\infty} \frac{|x|^n}{n!}\)

already converges for all real numbers.