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Chapter 9: Problem 5

Define infinite series and give an example.

Short Answer

Expert verified
Answer: An infinite series is the sum of the terms of an infinite sequence, represented as \(\sum_{n=1}^{\infty} a_n\). To calculate the sum of an infinite geometric series, use the formula \(S = \frac{a}{1-r}\), where S is the sum, a is the first term, and r is the common ratio. Note that not all infinite series have a finite sum, as some may diverge.

Step by step solution

01

1. Define Infinite Series

An infinite series is the sum of the terms of an infinite sequence. In other words, it is the result of adding up an infinite number of individual elements or terms. It can be represented as: \[ \sum_{n=1}^{\infty} a_n \] where \(a_n\) is the n-th term of the sequence. Keep in mind that not all infinite series have a finite sum; some of them may diverge, which means their sum approaches infinity or does not settle to a fixed value.
02

2. Example of an Infinite Series

The most common example of an infinite series comes from the geometric series. Let's use the following geometric series as an example: \[ \sum_{n=1}^{\infty} ar^{n-1} \] where, a is the first term of the series, r is the common ratio, and n is the position of the term in the series. For our example, let's take \(a = 1\) and \(r = \frac{1}{2}\).
03

3. Write Out the Terms of the Series

Now, let's write out the terms of our example series and get a better understanding of it: \[ S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \] This infinite series is the sum of terms that keep on getting halved. We can notice that this series will get closer and closer to 2 but never actually reach it, as it's an infinite series.
04

4. Compute the Sum of the Infinite Series

To find the sum of an infinite geometric series, we use the formula: \[ S = \frac{a}{1-r} \] where S is the sum, a is the first term, and r is the common ratio. In our example, a = 1 and r = 1/2. Plugging these values into the formula: \[S = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2\] So, the sum of our example infinite series is 2. In conclusion, we have defined an infinite series as the sum of the terms of an infinite sequence and used the geometric series as an example. We wrote out its terms and found its sum using the corresponding formula. Our example infinite series converges to a finite sum of 2.

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Most popular questions from this chapter

Showing that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} \operatorname{In} 1734,\) Leonhard Euler informally proved that \(\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6} .\) An elegant proof is outlined here that uses the inequality $$ \cot ^{2} x<\frac{1}{x^{2}}<1+\cot ^{2} x\left(\text { provided that } 0

Find the limit of the sequence $$\left\\{a_{n}\right\\}_{n=2}^{\infty}=\left\\{\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right) \cdots\left(1-\frac{1}{n}\right)\right\\}.$$

Determine whether the following statements are true and give an explanation or counterexample. a. A series that converges must converge absolutely. b. A series that converges absolutely must converge. c. A series that converges conditionally must converge. d. If \(\sum a_{k}\) diverges, then \(\Sigma\left|a_{k}\right|\) diverges. e. If \(\sum a_{k}^{2}\) converges, then \(\sum a_{k}\) converges. f. If \(a_{k}>0\) and \(\sum a_{k}\) converges, then \(\Sigma a_{k}^{2}\) converges. g. If \(\Sigma a_{k}\) converges conditionally, then \(\Sigma\left|a_{k}\right|\) diverges.

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. $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$

Determine whether the following series converge or diverge. $$\sum_{k=0}^{\infty} \frac{10}{k^{2}+9}$$
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