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Chapter 9: Problem 24

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{\ln (1 / n)}{n}\right\\}$$

Short Answer

Expert verified
Answer: The limit of the sequence as \(n\) approaches infinity is \(0\).

Step by step solution

01

Rewrite the sequence with limit notation

We want to find the limit as \(n\) approaches infinity. So, we write: $$\lim_{n\to\infty} \frac{\ln\left(\frac{1}{n}\right)}{n}$$
02

Express the logarithm as a natural log

Recall that \(\ln\left(\frac{1}{n}\right)\) is the natural logarithm of \(\frac{1}{n}\), which can also be written as \(-\ln(n)\). So, the limit becomes: $$\lim_{n\to\infty} \frac{-\ln(n)}{n}$$
03

Check L'Hopital's rule conditions

We can see that both the numerator and denominator tend to infinity as \(n\) approaches infinity. L'Hopital's rule can be applied when the limit of both the numerator and denominator is either zero or infinity. Since we have infinity in this case, we can proceed.
04

Apply L'Hopital's rule

Differentiate the numerator and denominator with respect to \(n\): $$\frac{d}{dn}(-\ln(n)) = -\frac{1}{n}$$ $$\frac{d}{dn}(n) = 1$$ Now, rewrite the limit with the new numerator and denominator: $$\lim_{n\to\infty} \frac{-\frac{1}{n}}{1}$$
05

Evaluate the limit

To evaluate the limit, we divide the numerator by the denominator as \(n\) approaches infinity: $$\lim_{n\to\infty} \left(-\frac{1}{n}\right) = 0$$ Thus, the limit of the sequence \(\left\\{\frac{\ln (1 / n)}{n}\right\\}\) as \(n\) approaches infinity is \(0\).

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Most popular questions from this chapter

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. $$\sum_{k=1}^{\infty} 3^{-k}$$

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 \(10 ?\)

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty}\left(-\frac{1}{3}\right)^{k}$$

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