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Chapter 8: Problem 83

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

Short Answer

Expert verified
Answer: The power series converges for values of \(x\) in the interval \([0,1)\).

Step by step solution

01

Identify the general term of the series

The general term of the given series is \(a_k = x^k\).
02

Write down the Ratio Test expression

Using the definition of the Ratio Test, find the expression as follows: $$\lim_{k \to \infty} \left|\frac{a_{k+1}}{a_k}\right| = \lim_{k \to \infty} \left|\frac{x^{k+1}}{x^k}\right|$$
03

Simplify the expression

Simplify the expression from step 2 to get: $$\lim_{k \to \infty} \left|\frac{x^{k+1}}{x^k}\right| = \lim_{k \to \infty} |x|$$
04

Determine convergence

In order for the series to converge, the Ratio Test requires the following condition to be met: $$\lim_{k \to \infty} |x| < 1$$ Since the limit is just a constant value \(|x|\), the condition is the same as requiring \(|x| < 1\). We are given that \(x \geq 0\), so we can write the condition as: $$0 \leq x < 1$$
05

State the interval of convergence

Based on our analysis using the Ratio Test, the power series \(\sum_{k=1}^{\infty} x^{k}\) converges for values of \(x\) in the interval \([0,1)\).

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Key Concepts

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

Power Series
A power series is a type of infinite series where each term consists of a variable raised to a power and multiplied by a coefficient. In mathematical terms, a power series can be represented as:
  • \( \sum_{k=0}^{\infty} a_k x^k \)
Here, \(a_k\) represents the coefficients, and \(x\) is the variable term. Power series are pivotal in mathematics because they allow functions to be represented as sums of infinitely many terms. This makes it easier to perform calculations or approximations in calculus.

When handling power series, we often want to know when they will converge, meaning the sum approaches a finite limit rather than becoming infinitely large. Determining convergence depends on both the values of \(x\) and the coefficients \(a_k\). Techniques such as the Ratio Test help us establish the intervals of \(x\) for which a given power series converges.
Convergence
Convergence in the context of series, including power series, refers to the property where the sum of the series approaches a limit. If a series is converging, its terms get closer to a particular value, making it very important to ascertain if a given power series converges for particular values of \(x\).

The Ratio Test is a common tool for testing convergence of a series. It involves evaluating:
  • \( \lim_{k \to \infty} \left| \frac{a_{k+1}}{a_k} \right| \)
Where \(a_k\) is the general term of the series. If the limit is less than 1, the series converges absolutely. If greater than 1, it diverges. If it equals 1, the test is inconclusive.

In our example, the power series \( \sum_{k=1}^{\infty} x^k \) converges for \(0 \leq x < 1\). This means that within this interval, the infinite series adds up to a finite number.
Infinite Series
An infinite series is essentially a sum of an infinite sequence of terms. Unlike finite sums, which have a definite number of terms, infinite series continue indefinitely. When dealing with infinite series, it鈥檚 crucial to determine if these sums converge to a finite value.

The convergence of an infinite series depends on the nature of its terms. The general form for an infinite series is:
  • \( \sum_{k=0}^{\infty} a_k \)
Where \(a_k\) represents each term in the series. For practical applications, especially in mathematics and physics, knowing whether an infinite series converges is essential as it informs us whether the sum is meaningful or not.

In the case of power series like \( \sum_{k=1}^{\infty} x^k \), using convergence tests such as the Ratio Test helps identify which values鈥攖ypically as intervals鈥攁llow the infinite series to yield a finite sum.

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Most popular questions from this chapter

A fishery manager knows that her fish population naturally increases at a rate of \(1.5 \%\) per month, while 80 fish are harvested each month. Let \(F_{n}\) be the fish population after the \(n\) th month, where \(F_{0}=4000\) fish. a. Write out the first five terms of the sequence \(\left\\{F_{n}\right\\}\). b. Find a recurrence relation that generates the sequence \(\left\\{F_{n}\right\\}\). c. Does the fish population decrease or increase in the long run? d. Determine whether the fish population decreases or increases in the long run if the initial population is 5500 fish. e. Determine the initial fish population \(F_{0}\) below which the population decreases.

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=e^{n / 2} \text { and } b_{n}=n^{5}, n \geq 2$$

A fallacy Explain the fallacy in the following argument. Let \(x=1+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\cdots\) and \(y=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\cdots \cdot\) It follows that \(2 y=x+y\) which implies that \(x=y .\) On the other hand, $$ x-y=\underbrace{\left(1-\frac{1}{2}\right)}_{>0}+\underbrace{\left(\frac{1}{3}-\frac{1}{4}\right)}_{>0}+\underbrace{\left(\frac{1}{5}-\frac{1}{6}\right)}_{>0}+\cdots>0 $$ is a sum of positive terms, so \(x>y .\) Therefore, we have shown that \(x=y\) and \(x>y\)

The Fibonacci sequence \(\\{1,1,2,3,5,8,13, \ldots\\}\) is generated by the recurrence relation \(f_{n+1}=f_{n}+f_{n-1},\) for \(n=1,2,3, \ldots,\) where \(f_{0}=1, f_{1}=1\). a. It can be shown that the sequence of ratios of successive terms of the sequence \(\left\\{\frac{f_{n+1}}{f_{n}}\right\\}\) has a limit \(\varphi .\) Divide both sides of the recurrence relation by \(f_{n},\) take the limit as \(n \rightarrow \infty,\) and show that \(\varphi=\lim _{n \rightarrow \infty} \frac{f_{n+1}}{f_{n}}=\frac{1+\sqrt{5}}{2} \approx 1.618\). b. Show that \(\lim _{n \rightarrow \infty} \frac{f_{n-1}}{f_{n+1}}=1-\frac{1}{\varphi} \approx 0.382\). c. Now consider the harmonic series and group terms as follows: $$\sum_{k=1}^{\infty} \frac{1}{k}=1+\frac{1}{2}+\frac{1}{3}+\left(\frac{1}{4}+\frac{1}{5}\right)+\left(\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)$$ $$+\left(\frac{1}{9}+\cdots+\frac{1}{13}\right)+\cdots$$ With the Fibonacci sequence in mind, show that $$\sum_{k=1}^{\infty} \frac{1}{k} \geq 1+\frac{1}{2}+\frac{1}{3}+\frac{2}{5}+\frac{3}{8}+\frac{5}{13}+\cdots=1+\sum_{k=1}^{\infty} \frac{f_{k-1}}{f_{k+1}}.$$ d. Use part (b) to conclude that the harmonic series diverges. (Source: The College Mathematics Journal, 43, May 2012)

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. Jack took a \(200-\mathrm{mg}\) dose of a painkiller at midnight. Every hour, \(5 \%\) of the drug is washed out of his bloodstream. Let \(d_{n}\) be the amount of drug in Jack's blood \(n\) hours after the drug was taken, where \(d_{0}=200 \mathrm{mg}\)
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