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Chapter 9: Problem 74

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{n}{n^{2}+1}=0$$

Short Answer

Expert verified
Answer: The limit of the sequence is 0.

Step by step solution

01

Recall the formal definition of the limit of a sequence

The formal definition of the limit of a sequence says that a sequence \((a_n)\) has a limit L as n approaches infinity if, for every positive real number 蔚, there exists a positive integer N such that for all n greater than N, the absolute value of the difference between \(a_n\) and L is less than 蔚: $$\forall 蔚 > 0, \exists N > 0 \text{ such that } \forall n > N, |a_n - L| < 蔚$$
02

Apply the formal definition to the given exercise

In this problem, we are given the sequence \(a_n = \frac{n}{n^2 + 1}\) and the limit L = 0. We need to show that for every 蔚 > 0, there is a positive integer N with the following property: $$\forall n > N, \left|\frac{n}{n^2 + 1} - 0\right| < 蔚$$
03

Simplify the inequality

We can simplify the inequality by removing the subtraction by zero and taking the absolute value of the function: $$\frac{n}{n^2 + 1} < 蔚$$
04

Solve for N in terms of 蔚

In order to find the N that satisfies the inequality for a given 蔚, we can rewrite the inequality as follows: $$n < 蔚(n^2 + 1)$$ Now, we can try to bound the inequality from above: $$n < 蔚(n^2 + 1) < 蔚n^2$$ Divide both sides of the inequality by 蔚: $$\frac{n}{蔚} < n^2$$ Now, to find the N in terms of 蔚, we can take the square root of both sides: $$\sqrt{\frac{n}{蔚}} < n$$ Let N be the smallest integer greater than \(\sqrt{\frac{n}{蔚}}\). This means that for \(\forall n > N\), the inequality \(\frac{n}{n^2 + 1} < 蔚\) holds. Therefore, the proof is complete and the limit of the sequence is 0: $$\lim _{n \rightarrow \infty} \frac{n}{n^{2}+1}=0$$

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Most popular questions from this chapter

Suppose a ball is thrown upward to a height of \(h_{0}\) meters. Each time the ball bounces, it rebounds to a fraction r of its previous height. Let \(h_{n}\) be the height after the nth bounce and let \(S_{n}\) be the total distance the ball has traveled at the moment of the nth bounce. a. Find the first four terms of the sequence \(\left\\{S_{n}\right\\}\) b. Make a table of 20 terms of the sequence \(\left\\{S_{n}\right\\}\) and determine a plausible value for the limit of \(\left\\{S_{n}\right\\}.\) $$h_{0}=20, r=0.5$$

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

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k}}{k^{2 / 3}}$$

Determine whether the following series converge absolutely or conditionally, or diverge. $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k^{3 / 2}}$$

Consider the geometric series $$S=\sum_{k=0}^{\infty} r^{k}$$ which has the value \(1 /(1-r)\) provided \(|r|<1 .\) Let \(S_{n}=\sum_{k=0}^{n-1} r^{k}=\frac{1-r^{n}}{1-r}\) be the sum of the first \(n\) terms. The remainder \(R_{n}\) is the error in approximating \(S\) by \(S_{n} .\) Show that $$R_{n}=\left|S-S_{n}\right|=\left|\frac{r^{n}}{1-r}\right|$$
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