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Chapter 9: Problem 19

Evaluate the geometric series or state that it diverges. $$\sum_{k=0}^{\infty}\left(\frac{1}{4}\right)^{k}$$

Short Answer

Expert verified
Answer: The sum of the given geometric series is 4/3.

Step by step solution

01

Identify the common ratio

The common ratio in the given geometric series is \(\frac{1}{4}\). Since the common ratio is between -1 and 1, we can conclude that the series converges.
02

Use the formula to evaluate the sum

The formula to find the sum of an infinite geometric series is: $$S = \frac{a_1}{1 - r}$$ Here, \(a_1\) is the first term of the series and \(r\) is the common ratio. In the given series, we have \(a_1 = \left(\frac{1}{4}\right)^0 = 1\) and \(r = \frac{1}{4}\).
03

Calculate the sum

Now, using the formula, we can find the sum of the series: $$S = \frac{1}{1 - \frac{1}{4}}$$
04

Simplify the expression

Now, we will simplify the expression for the sum: $$S = \frac{1}{\frac{3}{4}} = \frac{4}{3}$$
05

Conclusion

The sum of the given geometric series is \(\frac{4}{3}\).

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Key Concepts

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

Infinite Series
An infinite series is a sum of an infinite sequence of numbers. Rather than stopping at a finite number, it continues without end. In our example, \(\sum_{k=0}^{\infty}\left(\frac{1}{4}\right)^{k}\), this series includes all terms starting from \(k=0\) to infinity. Such series can either converge to a specific value, or diverge, meaning they don't settle on a specific number. Understanding whether an infinite series converges or diverges is crucial in determining its sum.
Convergence
Convergence is all about whether an infinite series approaches a finite value as more terms are added. A series like \(\sum_{k=0}^{\infty}\left(\frac{1}{4}\right)^{k}\) converges if it steadily gets closer to a single number. This happens when the common ratio, \(r\), is between -1 and 1. If the series converges, we can calculate its sum using a specific formula. But if it diverges, it doesn't reach a finite value, so we can't determine a meaningful sum.
Common Ratio
The common ratio is the number each term is multiplied by to get the next term in a geometric series. In our exercise, the common ratio is \(\frac{1}{4}\). This ratio is crucial because it tells us a lot about the series:
  • If \(|r| < 1\), the series converges.
  • If \(|r| \geq 1\), the series diverges.
For \(\sum_{k=0}^{\infty}\left(\frac{1}{4}\right)^{k}\), since \(\frac{1}{4}\) is between -1 and 1, the series converges.
Sum Formula
When dealing with a convergent infinite geometric series, the sum can be found using the formula:\[S = \frac{a_1}{1 - r}\]Here, \(a_1\) is the first term, and \(r\) is the common ratio.
  • In our series, \(a_1 = 1\) because \(\left(\frac{1}{4}\right)^0 = 1\).
  • The common ratio \(r\) is \(\frac{1}{4}\).
Plugging these into the formula:\[S = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}\]This tells us the sum of the series is \(\frac{4}{3}\). Understanding this formula helps quickly find sums for similar series.

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Most popular questions from this chapter

For what values of \(x\) does the geometric series $$f(x)=\sum_{k=0}^{\infty}\left(\frac{1}{1+x}\right)^{k}$$ converge? Solve \(f(x)=3\)

Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year \(1984 .\) Assume the CPI has increased by an average of \(3 \%\) per year since \(1984 .\) Let \(c_{n}\) be the CPI \(n\) years after \(1984,\) where \(c_{0}=100.\)

Convergence parameter Find the values of the parameter \(p>0\) for which the following series converge. $$\sum_{k=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdots(2 k-1)}{p^{k} k !}$$

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 \(0.01.\)

a. Sketch the function \(f(x)=1 / x\) on the interval \([1, n+1]\) where \(n\) is a positive integer. Use this graph to verify that $$ \ln (n+1)<1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}<1+\ln n $$ b. Let \(S_{n}\) be the sum of the first \(n\) terms of the harmonic series, so part (a) says \(\ln (n+1)0,\) for \(n=1,2,3, \ldots\) c. Using a figure similar to that used in part (a), show that $$ \frac{1}{n+1}>\ln (n+2)-\ln (n+1) $$ d. Use parts (a) and (c) to show that \(\left\\{E_{n}\right\\}\) is an increasing sequence \(\left(E_{n+1}>E_{n}\right)\) e. Use part (a) to show that \(\left\\{E_{n}\right\\}\) is bounded above by 1 f. Conclude from parts (d) and (e) that \(\left\\{E_{n}\right\\}\) has a limit less than or equal to \(1 .\) This limit is known as Euler's constant and is denoted \(\gamma\) (the Greek lowercase gamma). g. By computing terms of \(\left\\{E_{n}\right\\},\) estimate the value of \(\gamma\) and compare it to the value \(\gamma \approx 0.5772 .\) (It has been conjectured, but not proved, that \(\gamma\) is irrational.) h. The preceding arguments show that the sum of the first \(n\) terms of the harmonic series satisfy \(S_{n} \approx 0.5772+\ln (n+1)\) How many terms must be summed for the sum to exceed \(10 ?\)
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