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Chapter 8: Problem 73

Use the formal definition of the limit of a sequence to prove the following limits. $$\lim _{n \rightarrow \infty} \frac{c n}{b n+1}=\frac{c}{b}, \text { for real numbers } b > 0 \text { and } c > 0$$

Short Answer

Expert verified
Question: Prove that the limit of the sequence $$\frac{cn}{bn+1}$$ as n approaches infinity is $$\frac{c}{b}$$ using the formal definition of a limit. Answer: We can prove the limit by selecting an epsilon (蔚) and finding a positive integer (N) such that for all integers n > N, the absolute difference between the sequence and the expected limit is less than 蔚. We manipulate the inequality $$\left|\frac{cn}{bn+1} - \frac{c}{b}\right| < \epsilon$$ and find a suitable N by taking the smallest integer value greater than the obtained inequality $$N = \left\lceil\frac{c}{b^2\epsilon} - \frac{1}{b}\right\rceil$$. This confirms the limit $$\lim _{n \rightarrow \infty} \frac{c n}{b n+1}=\frac{c}{b}$$.

Step by step solution

01

Set up the inequality

We want to find a positive integer N such that for all integers n > N, the following inequality holds true: $$\left|\frac{cn}{bn+1} - \frac{c}{b}\right| < \epsilon$$
02

Manipulate the inequality

Combine the fractions to work with a single fraction. $$\left|\frac{bcn - c(bn+1)}{b(bn+1)}\right| < \epsilon$$ Simplify the numerator and factor c. $$\left|\frac{c(-1)}{b(bn+1)}\right| = \frac{c}{b(bn+1)} < \epsilon$$ As c, b > 0, the fraction will always be positive, so we can drop the absolute value. $$\frac{c}{b(bn+1)} < \epsilon$$
03

Set up an inequality for N

To make the left side of the inequality less than 蔚, we can use the following observation: $$\frac{c}{b(bn+1)} < \frac{c}{b(bN)}$$ Therefore, $$\frac{c}{b(bN)} < \epsilon$$ From this, we can solve for N. $$N > \frac{c}{b^2\epsilon} - \frac{1}{b}$$
04

Define N

We can take the smallest integer value greater than the obtained inequality for a valid N: $$N = \left\lceil\frac{c}{b^2\epsilon} - \frac{1}{b}\right\rceil$$
05

Confirm the result

For any 蔚, we have now found an integer N such that if n > N, then $$\left|\frac{cn}{bn+1} - \frac{c}{b}\right| < \epsilon$$ This confirms the limit: $$\lim _{n \rightarrow \infty} \frac{c n}{b n+1}=\frac{c}{b}$$

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Most popular questions from this chapter

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 magnitude of the remainder \(R_{n}\) is the error in approximating \(S\) by \(S_{n} .\) Show that $$ R_{n}=S-S_{n}=\frac{r^{n}}{1-r} $$

For a positive real number \(p,\) the tower of exponents \(p^{p^{p}}\) continues indefinitely and the expression is ambiguous. The tower could be built from the top as the limit of the sequence \(\left\\{p^{p},\left(p^{p}\right)^{p},\left(\left(p^{p}\right)^{p}\right)^{p}, \ldots .\right\\},\) in which case the sequence is defined recursively as \(a_{n+1}=a_{n}^{p}(\text { building from the top })\) where \(a_{1}=p^{p} .\) The tower could also be built from the bottom as the limit of the sequence \(\left\\{p^{p}, p^{\left(p^{p}\right)}, p^{\left(p^{(i)}\right)}, \ldots .\right\\},\) in which case the sequence is defined recursively as \(a_{n+1}=p^{a_{n}}(\text { building from the bottom })\) where again \(a_{1}=p^{p}\). a. Estimate the value of the tower with \(p=0.5\) by building from the top. That is, use tables to estimate the limit of the sequence defined recursively by (1) with \(p=0.5 .\) Estimate the maximum value of \(p > 0\) for which the sequence has a limit. b. Estimate the value of the tower with \(p=1.2\) by building from the bottom. That is, use tables to estimate the limit of the sequence defined recursively by (2) with \(p=1.2 .\) Estimate the maximum value of \(p > 1\) for which the sequence has a limit.

A ball is thrown upward to a height of \(h_{0}\) meters. After each bounce, the ball 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 \(\left\\{S_{n}\right\\}\) $$h_{0}=20, r=0.5$$

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

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