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Chapter 8: Problem 7

Compare the growth rates of \(\left\\{n^{100}\right\\}\) and \(\left\\{e^{n / 100}\right\\}\) as \(n \rightarrow \infty\).

Short Answer

Expert verified
Answer: The function \(e^{\frac{n}{100}}\) grows faster than \(n^{100}\) as \(n\) approaches infinity.

Step by step solution

01

Write the Ratio

Write the ratio of the two functions: \(\frac{n^{100}}{e^{\frac{n}{100}}}\).
02

Evaluate the limit

Evaluate the limit of the ratio as \(n\) approaches infinity: $$\lim_{n \to \infty} \frac{n^{100}}{e^{\frac{n}{100}}}$$
03

Use L'H么pital's Rule

Since we have an indeterminate form (\(\frac{\infty}{\infty}\)), we can apply L'H么pital's Rule. Differentiate the numerator and denominator with respect to \(n\): $$\frac{d}{dn}\left(n^{100}\right) = 100 n^{99}$$ $$\frac{d}{dn}\left(e^{\frac{n}{100}}\right) = \frac{1}{100} e^{\frac{n}{100}}$$ Now, compute the limit of the ratio of the derivatives: $$\lim_{n \to \infty} \frac{100 n^{99}}{\frac{1}{100} e^{\frac{n}{100}}}$$
04

Simplify the expression

Simplify the expression by multiplying both the numerator and the denominator by 100: $$\lim_{n \to \infty} \frac{10000 n^{99}}{e^{\frac{n}{100}}}$$
05

Apply L'H么pital's Rule repeatedly

Since we still have an indeterminate form (\(\frac{\infty}{\infty}\)), continue to apply L'H么pital's Rule repeatedly until we obtain either a finite nonzero limit or a determinate form. After 99 more applications, we get: $$\lim_{n \to \infty} \frac{100^{100} \cdot 100!}{e^{\frac{n}{100}}}$$
06

Evaluate the final limit

As \(n\) approaches infinity, the exponential term in the denominator \(e^{\frac{n}{100}}\) will dominate, and the limit becomes: $$\lim_{n \to \infty} \frac{100^{100} \cdot 100!}{e^{\frac{n}{100}}} = 0$$
07

Interpret the result

Since the limit is 0, we can conclude that the growth rate of \(n^{100}\) is slower than the growth rate of \(e^{\frac{n}{100}}\) as \(n\) approaches infinity.

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Most popular questions from this chapter

In the following exercises, two sequences are given, one of which initially has smaller values, but eventually "overtakes" the other sequence. Find the sequence with the larger growth rate and the value of \(n\) at which it overtakes the other sequence. $$a_{n}=\sqrt{n} \text { and } b_{n}=2 \ln n, n \geq 3$$

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

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\frac{6^{n}+3^{n}}{6^{n}+n^{100}}$$

Use Theorem 8.6 to find the limit of the following sequences or state that they diverge. $$\left\\{\frac{n^{10}}{\ln ^{1000} n}\right\\}$$

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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