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Chapter 8: Problem 82

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

Short Answer

Expert verified
Answer: The series converges for all \(x \geq 0\).

Step by step solution

01

Identify the terms of the series

The series is given by: $$\sum_{k=1}^{\infty} \frac{x^{k}}{k !}$$ The terms of the series are: $$a_k = \frac{x^{k}}{k !}$$
02

Apply the Ratio Test

We will calculate the limit of the ratio of consecutive terms as \(k \to \infty\): $$\lim_{k \to \infty} \frac{a_{k+1}}{a_k} = \lim_{k \to \infty} \frac{\frac{x^{k+1}}{(k+1) !}}{\frac{x^{k}}{k !}}$$
03

Simplify the expression

To simplify the expression, we perform the following operations: $$\lim_{k \to \infty} \frac{(k!)x^{k+1}}{(k+1)!x^k} = \lim_{k \to \infty} \frac{x}{k+1}$$
04

Determine the limit

We can see that the limit is: $$\lim_{k \to \infty} \frac{x}{k+1} = 0$$
05

Convergence interval based on the Ratio Test

Since the limit is 0, which is less than 1, the series converges for all non-negative values of \(x\), i.e., \(x\geq 0\). The series converges for all \(x \geq 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Power Series
Understanding the concept of a power series is essential for exploring functions in calculus. It represents a function as an infinite sum of terms that are powers of a variable, usually denoted by 'x'. These terms are multiplied by coefficients that can be constants or depend on the index of the term. The general form of a power series is \[\sum_{n=0}^{fty} c_n(x-a)^n\] where \(c_n\) are the coefficients, \(x\) is the variable, \(a\) is the center of the series, and \(n\) indicates the nth term.

Series Convergence
A critical aspect of working with series is determining whether they converge or diverge; that is, if the sum of their infinite terms approaches a finite number (convergence) or not (divergence). The Ratio Test is a common method for determining a series' behavior. This test involves taking the limit of the ratio of consecutive terms (as seen with \(\frac{a_{k+1}}{a_k}\). If the limit is less than 1, the series is said to converge. Convergence is key to ensuring that the series sums up to a finite, well-defined value, making it possible to use power series for practical computations.

Factorial Notation in Series
Factorial notation is frequently encountered in series, particularly those involving exponential expressions and combinations. The factorial of a positive integer \(k\), denoted as \(k!\), is the product of all positive integers up to \(k\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). In series, factorials provide a way to describe rapidly growing or shrinking sequence elements. They are particularly useful in power series expressions of exponential functions, such as the one described in our exercise \(\sum_{k=1}^{fty} \frac{x^{k}}{k !}\), which converges for all non-negative values of \(x\).

Limit of a Sequence
In mathematics, the concept of the limit of a sequence is a fundamental part of analysis, focusing on the behavior of sequences as they progress towards infinity. A sequence \(a_n\) has a limit \(L\) if, as \(n\) gets larger and larger, the sequence's terms get arbitrarily close to \(L\). In the context of series and the Ratio Test, we look for the limit of the ratio of consecutive terms to determine convergence. If this ratio tends towards zero, as with our exemplified series \(\lim_{k \to fty} \frac{x}{k+1} = 0\), the series is guaranteed to converge, which plays a crucial role in evaluating the behavior of power series across different values of \(x\).

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Most popular questions from this chapter

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty} \frac{\pi^{k}}{e^{k+1}}$$

An insulated window consists of two parallel panes of glass with a small spacing between them. Suppose that each pane reflects a fraction \(p\) of the incoming light and transmits the remaining light. Considering all reflections of light between the panes, what fraction of the incoming light is ultimately transmitted by the window? Assume the amount of incoming light is 1.

$$\text {Evaluate each series or state that it diverges.}$$ $$\sum_{k=1}^{\infty}\left(\sin ^{-1}(1 / k)-\sin ^{-1}(1 /(k+1))\right)$$

Consider the following infinite series. a. Write out the first four terms of the sequence of partial sums. b. Estimate the limit of \(\left\\{S_{n}\right\\}\) or state that it does not exist. $$\sum_{k=1}^{\infty}(-1)^{k} k$$

Repeated square roots Consider the expression \(\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}},\) where the process continues indefinitely. a. Show that this expression can be built in steps using the recurrence relation \(a_{0}=1, a_{n+1}=\sqrt{1+a_{n}}\), for \(n=0,1,2,3, \ldots . .\) Explain why the value of the expression can be interpreted as \(\lim _{n \rightarrow \infty} a_{n},\) provided the limit exists. b. Evaluate the first five terms of the sequence \(\left\\{a_{n}\right\\}\) c. Estimate the limit of the sequence. Compare your estimate with \((1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Assuming the limit exists, use the method of Example 5 to determine the limit exactly. e. Repeat the preceding analysis for the expression \(\sqrt{p+\sqrt{p+\sqrt{p+\sqrt{p+\cdots}}},}\) where \(p>0 .\) Make a table showing the approximate value of this expression for various values of \(p .\) Does the expression seem to have a limit for all positive values of \(p ?\)
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