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Chapter 9: Problem 23

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{n}{e^{n}+3 n}\right\\}$$

Short Answer

Expert verified
Answer: The limit of the sequence as n tends to infinity is 0.

Step by step solution

01

Identify the form of the sequence

First, let's try to determine the form of the given sequence as n tends to infinity: $$\lim_{n\to\infty} \frac{n}{e^{n}+3 n}$$ As n tends to infinity, the numerator goes to infinity, while the denominator also goes to infinity due to the exponential term. Therefore, we have an indeterminate form of the type \(鈭�/鈭瀄).
02

Apply L'Hopital's Rule

Since we have an indeterminate form of the type \(鈭�/鈭瀄), we can apply L'Hopital's Rule. This requires taking the derivative of the numerator (n) and the denominator (\(e^{n}+3n\)) with respect to n and then finding the limit of the new sequence as n tends to infinity: $$\lim_{n\to\infty} \frac{\frac{d}{dn}(n)}{\frac{d}{dn}(e^{n}+3 n)}$$
03

Calculate the Derivatives

Now we will find the derivatives of the numerator and the denominator with respect to n: $$\frac{d}{dn}(n) = 1$$ and $$\frac{d}{dn}(e^{n}+3 n) = e^{n} + 3$$
04

Find the Limit

Replace the numerator and the denominator with their derivatives: $$\lim_{n\to\infty} \frac{1}{e^{n} + 3}$$ As n tends to infinity, the term \(e^{n}\) grows rapidly and dominates the denominator, making the value of the entire denominator increase without bound. Therefore, the limit evaluates to: $$\lim_{n\to\infty} \frac{1}{e^{n} + 3} = \frac{1}{\infty} = 0$$
05

State the Result

The limit of the given sequence as n tends to infinity is 0: $$\lim_{n\to\infty} \frac{n}{e^{n}+3 n} = 0$$

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Given any infinite series \(\Sigma a_{k},\) let \(N(r)\) be the number of terms of the series that must be summed to guarantee that the remainder is less than \(10^{-r}\), where \(r\) is a positive integer. a. Graph the function \(N(r)\) for the three alternating \(p\) -series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{p}},\) for \(p=1,2,\) and \(3 .\) Compare the three graphs and discuss what they mean about the rates of convergence of the three series. b. Carry out the procedure of part (a) for the series \(\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k !}\) and compare the rates of convergence of all four series.
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