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Chapter 9: Problem 5

For what values of \(r\) does the sequence \(\left\\{r^{n}\right\\}\) converge? Diverge?

Short Answer

Expert verified
The sequence \(r^n\) converges for values of \(0 \leq r \leq 1\) and diverges for values of \(r < 0\) or \(r > 1\).

Step by step solution

01

Analyze the behavior of the sequence when \(r>1\)

For \(r>1\), let's consider the limit as \(n\) goes to infinity: \(\lim_{n \rightarrow \infty} r^n\) Since \(r > 1\), as \(n\) increases, the value of \(r^n\) becomes larger and larger. Thus, this limit doesn't exist as it goes to infinity: \(\lim_{n \rightarrow \infty} r^n = \infty\) So, for values of \(r>1\), the sequence \(\left\\{r^{n}\right\\}\) diverges.
02

Analyze the behavior of the sequence when \(0

For \(0 < r < 1\), let's consider the limit as \(n\) goes to infinity: \(\lim_{n \rightarrow \infty} r^n\) Since \(0 < r < 1\), as \(n\) increases, the value of \(r^n\) becomes smaller and smaller. This means that the limit of the sequence as \(n\) goes to infinity is zero: \(\lim_{n \rightarrow \infty} r^n = 0\) So, for values of \(0<r<1\), the sequence \(\left\\{r^{n}\right\\}\) converges to 0.
03

Analyze the behavior of the sequence when \(r

For \(r < 0\), the sequence will alternate signs having positive and negative values, since odd powers of a negative number will be negative and even powers of a negative number will be positive. As absolute value of \(r^n\) behaves same as in the case when r is positive, limit as n goes to infinity doesn't converge to any single value because of oscillations. So, for values of \(r<0\), the sequence \(\left\\{r^{n}\right\\}\) diverges.
04

Analyze the behavior of the sequence when \(r=0\) or \(r=1\)

For \(r=0\), the sequence will always be: \(0, 0, 0, \dots\) As the terms of the sequence don't change with \(n\), the sequence converges to 0. For \(r=1\), the sequence becomes: \(1, 1, 1, \dots\) As all terms in the sequence are equal to 1, the sequence converges to 1. So, for values of \(r=0\) or \(r=1\), the sequence \(\left\\{r^{n}\right\\}\) converges.
05

Conclusion

The sequence \(\left\\{r^{n}\right\\}\) converges for values of \(0 \leq r \leq 1\) and diverges for values of \(r < 0\) or \(r > 1\).

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Most popular questions from this chapter

The Riemann zeta function is the subject of extensive research and is associated with several renowned unsolved problems. It is defined by \(\zeta(x)=\sum_{k=1}^{\infty} \frac{1}{k^{x}}\). When \(x\) is a real number, the zeta function becomes a \(p\) -series. For even positive integers \(p,\) the value of \(\zeta(p)\) is known exactly. For example, $$ \sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}, \quad \sum_{k=1}^{\infty} \frac{1}{k^{4}}=\frac{\pi^{4}}{90}, \quad \text { and } \quad \sum_{k=1}^{\infty} \frac{1}{k^{6}}=\frac{\pi^{6}}{945}, \ldots $$ Use estimation techniques to approximate \(\zeta(3)\) and \(\zeta(5)\) (whose values are not known exactly) with a remainder less than \(10^{-3}\).

Suppose a function \(f\) is defined by the geometric series \(f(x)=\sum_{k=0}^{\infty}(-1)^{k} x^{k}\) a. Evaluate \(f(0), f(0.2), f(0.5), f(1),\) and \(f(1.5),\) if possible. b. What is the domain of \(f ?\)

Series of squares Prove that if \(\sum a_{k}\) is a convergent series of positive terms, then the series \(\Sigma a_{k}^{2}\) also converges.

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

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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