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Chapter 9: Problem 45

Use the properties of infinite series to evaluate the following series. $$\sum_{k=0}^{\infty}\left[3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right]$$

Short Answer

Expert verified
$$ Answer: The sum of the given infinite series is -2.

Step by step solution

01

Identify the two geometric series

The given series can be considered as a sum of two separate geometric series: $$\sum_{k=0}^{\infty}\left[3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right] = \sum_{k=0}^{\infty}3\left(\frac{2}{5}\right)^{k} - \sum_{k=0}^{\infty}2\left(\frac{5}{7}\right)^{k}.$$ The first geometric series is: $$\sum_{k=0}^{\infty}3\left(\frac{2}{5}\right)^{k}$$ with \(a_1 = 3\) and \(r_1 = \frac{2}{5}\). The second geometric series is: $$\sum_{k=0}^{\infty}2\left(\frac{5}{7}\right)^{k}$$ with \(a_2 = 2\) and \(r_2 = \frac{5}{7}\).
02

Determine the sum of each geometric series

For a convergent geometric series, the sum is given by the formula: \(S = \frac{a}{1-r}\), where \(a\) is the first term of the series and \(r\) is the common ratio. Since both \(r_1\) and \(r_2\) have an absolute value less than 1, both geometric series are convergent. For the first geometric series: $$S_1 = \frac{3}{1-\frac{2}{5}} = \frac{3}{\frac{3}{5}} = 5$$ For the second geometric series: $$S_2 = \frac{2}{1-\frac{5}{7}} = \frac{2}{\frac{2}{7}} = 7$$
03

Combine the sums of the geometric series

Now we can find the sum of the original series by subtracting the sum of the second geometric series from the sum of the first geometric series: $$\sum_{k=0}^{\infty}\left[3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right] = S_1 - S_2 = 5 - 7 = -2$$ Therefore, the overall sum of the given series is: $$\sum_{k=0}^{\infty}\left[3\left(\frac{2}{5}\right)^{k}-2\left(\frac{5}{7}\right)^{k}\right] = -2.$$

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Key Concepts

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

Infinite Series
When dealing with an infinite series, we address the sum of terms that continue indefinitely. A series is represented by a summation symbol and expresses the addition of an infinite number of terms. For instance, in the given exercise, the series runs from \(k=0\) to infinity. The terms within this series involve powers of a constant ratio \(r\), expressed as \(a_k(r)^k\), where \(a_k\) is a constant multiplier for each sequence.

Infinite series can either converge or diverge. Depending on the ratio of the terms as they progress to infinity, the series can reach a finite sum (converge) or not (diverge). It's fascinating how an infinite series like the sum in this exercise can have a relatively simple and finite result.
Series Convergence
For an infinite series to be useful, we often want it to "converge," meaning it approaches a specific value even as the number of terms becomes infinitely large. A geometric series is a common type of series where each term is a constant multiple \(r\) of the previous term.

To check for convergence in a geometric series, the absolute value of the ratio \(|r|\) must be less than 1. In the exercise, both series have ratios: \(r_1 = \frac{2}{5}\) and \(r_2 = \frac{5}{7}\), which are less than 1, confirming that they converge.
  • This means, despite the series having an infinite number of terms, they sum up to a finite number.
  • Convergence is crucial as it implies the series behaves in a predictable and measurable way.
Sum of a Series
The sum of a geometric series that converges can be calculated using a simple formula. If the first term of the series is \(a_1\) and the common ratio is \(r\), the sum \(S\) is given by \(S = \frac{a_1}{1-r}\) whenever \(|r| < 1\). This formula effectively summarizes infinitely many terms into a single finite result.

In the step by step solution, we see how the sum is calculated for both geometric series:
  • First series: With \(a_1 = 3\) and \(r_1 = \frac{2}{5}\), its sum \(S_1\) is 5.
  • Second series: With \(a_2 = 2\) and \(r_2 = \frac{5}{7}\), its sum \(S_2\) is 7.
These sums are then combined, letting us find the sum of the original series as \(-2\). Calculating the sum is not only about evaluating values but also about understanding how infinite elements can create a finite whole.

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Most popular questions from this chapter

A well-known method for approximating \(\sqrt{c}\) for positive real numbers \(c\) consists of the following recurrence relation (based on Newton's method). Let \(a_{0}=c\) and $$a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right), \quad \text { for } n=0,1,2,3, \dots$$ a. Use this recurrence relation to approximate \(\sqrt{10} .\) How many terms of the sequence are needed to approximate \(\sqrt{10}\) with an error less than \(0.01 ?\) How many terms of the sequence are needed to approximate \(\sqrt{10}\) with an error less than \(0.0001 ?\) (To compute the error, assume a calculator gives the exact value.) b. Use this recurrence relation to approximate \(\sqrt{c},\) for \(c=2\) \(3, \ldots, 10 .\) Make a table showing how many terms of the sequence are needed to approximate \(\sqrt{c}\) with an error less than \(0.01.\)

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 \(\lim _{n \rightarrow \infty} a_{n}\). 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 \(p ?\)

Two sine series Determine whether the following series converge. a. \(\sum_{k=1}^{\infty} \sin \frac{1}{k}\) b. \(\sum_{k=1}^{\infty} \frac{1}{k} \sin \frac{1}{k}\)

a. Consider the number 0.555555...., which can be viewed as the series \(5 \sum_{k=1}^{\infty} 10^{-k} .\) Evaluate the geometric series to obtain a rational value of \(0.555555 \ldots\) b. Consider the number \(0.54545454 \ldots,\) which can be represented by the series \(54 \sum_{k=1}^{\infty} 10^{-2 k} .\) Evaluate the geometric series to obtain a rational value of the number. c. Now generalize parts (a) and (b). Suppose you are given a number with a decimal expansion that repeats in cycles of length \(p,\) say, \(n_{1}, n_{2} \ldots \ldots, n_{p},\) where \(n_{1}, \ldots, n_{p}\) are integers between 0 and \(9 .\) Explain how to use geometric series to obtain a rational form of the number. d. Try the method of part (c) on the number \(0.123456789123456789 \ldots\) e. Prove that \(0 . \overline{9}=1\)

The famous Fibonacci sequence was proposed by Leonardo Pisano, also known as Fibonacci, in about A.D. 1200 as a model for the growth of rabbit populations. It is given by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1}\), for \(n=1,2,3, \ldots,\) where \(f_{0}=0, f_{1}=1 .\) Each term of the sequence is the sum of its two predecessors. a. Write out the first ten terms of the sequence. b. Is the sequence bounded? c. Estimate or determine \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}},\) the ratio of the successive terms of the sequence. Provide evidence that \(\varphi=(1+\sqrt{5}) / 2,\) a number known as the golden mean. d. Verify the remarkable result that $$f_{n}=\frac{1}{\sqrt{5}}\left(\varphi^{n}-(-1)^{n} \varphi^{-n}\right)$$
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