Question

A glimpse ahead to power series Use the Ratio Test to determine the values of \(x \geq 0\) for which each series converges. $$\sum_{k=1}^{\infty} \frac{x^{k}}{k !}$$

Short answer

Question: Determine the values of x greater than or equal to 0 for which the series converges: \(\sum_{k=1}^{\infty} \frac{x^{k}}{k!}\) Answer: The series converges for all values of x greater than or equal to 0.

Step-by-step solution

Write down the given series

The given series is: $$\sum_{k=1}^{\infty} \frac{x^{k}}{k!}$$

Calculate the ratio of consecutive terms

For the Ratio Test, we wish to find the limit of the ratio of consecutive terms: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_{k}}$$ In our series, we have \(a_{k}=\frac{x^{k}}{k!}\) and \(a_{k+1}=\frac{x^{k+1}}{(k+1)!}\). Divide \(a_{k+1}\) by \(a_{k}\): $$\frac{a_{k+1}}{a_{k}} = \frac{\frac{x^{k+1}}{(k+1)!}}{\frac{x^{k}}{k!}}$$

Simplify the fraction

Now, we will simplify the fraction by multiplying the numerator by the reciprocal of the denominator and simplify the powers of x and the factorials: $$\frac{a_{k+1}}{a_{k}} = \frac{x^{k+1}}{(k+1)!} \cdot \frac{k!}{x^{k}}$$ $$= \frac{x^{k+1}k!}{(k+1)!x^{k}}$$ $$= \frac{x^{k+1}}{x^{k}}\cdot \frac{k!}{(k+1)!}$$ $$= x \cdot \frac{1}{k+1}$$

Take the limit as k approaches infinity

Now, we take the limit of this expression as k approaches infinity: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_{k}} = \lim_{k \to \infty} x \cdot \frac{1}{k+1}$$ Since the term \(\frac{1}{k+1}\) approaches zero as k goes to infinity: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_{k}} = x \cdot 0$$ $$= 0$$

Apply the Ratio Test

Since the limit is less than 1: $$\lim_{k \to \infty} \frac{a_{k+1}}{a_{k}} = 0<1$$ Thus, the series converges for all values of x greater than or equal to 0.

Key concepts

power series

Power series are a type of infinite series that are very intriguing and useful in mathematics. They are defined as an infinite sum of terms in the form of \(a_k x^k\), where \(a_k\) represents the coefficients and \(x\) is a variable. Each term involves a power of \(x\), making it a powerful tool for expressing functions in the form of an infinite polynomial. This algebraic structure allows us to perform operations like differentiation and integration term by term.

• Power series converge within a certain radius around a center point. This range is called the "radius of convergence".
• The interval of convergence tells us the values of \(x\) for which the series will converge.

It is essential to establish where this series converges to ensure valid mathematical manipulation and representation.

factorials

Factorials are a fundamental concept in mathematics that often appear in power series and other mathematical expressions. The factorial of a non-negative integer \(n\), denoted as \(n!\), is the product of all positive integers less than or equal to \(n\).

• Factorials grow very rapidly with larger \(n\). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).
• Factorials are instrumental in combinatorics, which deals with counting, arrangements, and probability.

In the series mentioned above, factorials appear in the denominator of each term. This results in terms diminishing rapidly as \(k\) increases, often aiding in the convergence of series like the power series shown.

limit evaluation

Limit evaluation is a crucial concept in calculus that allows us to find the behavior of functions as they approach a particular point. In the context of the Ratio Test, we assess limits to determine the behavior of the ratio of consecutive terms in the series.

• The Ratio Test uses the limit of \( \frac{a_{k+1}}{a_k} \) to judge the convergence of series.
• If \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| < 1 \), the series converges.

In the exercise, calculating this limit involves breaking down and simplifying expressions involving \(x\) and factorials, which eventually showed the series converging for all non-negative values of \(x\).

series convergence

When we talk about series convergence, we are interested in whether the sum of terms in an infinite series will reach a finite value. The Ratio Test is one of the most effective methods to determine convergence, especially for series with factorials or exponential terms.

• Using the calculated limit, the Ratio Test confirms series convergence if the limit is less than 1.
• This method provides a straightforward approach to establishing whether the infinite process can settle at a specific sum.

In our original exercise, applying the Ratio Test resulted in a limit of 0, indicating the series converges for all non-negative \(x\). Understanding convergence is crucial for using power series safely in mathematical applications.