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Chapter 9: Problem 9

Find the limit of the following sequences or determine that the limit does not exist. $$\left\\{\frac{n^{3}}{n^{4}+1}\right\\}$$

Short Answer

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

Step by step solution

01

The numerator has a power of n^3, while the denominator has a power of n^4. The highest power in this sequence is n^4. #Step 2: Divide the numerator and the denominator by the highest power of n#

Divide both the numerator and the denominator by n^4 to simplify the expression and make the limit easier to determine: $$\frac{n^3}{n^4 + 1} = \frac{n^3/n^4}{(n^4 + 1)/n^4}$$ #Step 3: Simplify the expression#
02

Simplify the expression by canceling out the powers of n: $$\frac{n^3/n^4}{(n^4 + 1)/n^4} = \frac{1/n}{(1 + 1/n^4)}$$ #Step 4: Find the limit as n approaches infinity#

As n approaches infinity, both 1/n and 1/n^4 will approach 0: $$\lim_{n\to\infty}\frac{1/n}{(1 + 1/n^4)} = \frac{0}{(1+0)} = 0$$ The limit of the given sequence 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. $$\sum_{k=1}^{\infty} 9(0.1)^{k}$$

Evaluate the limit of the following sequences. $$a_{n}=\frac{75^{n-1}}{99^{n}}+\frac{5^{n} \sin n}{8^{n}}$$

Evaluate the limit of the following sequences. $$a_{n}=\frac{n^{8}+n^{7}}{n^{7}+n^{8} \ln n}$$

Determine whether the following series converge or diverge. $$\sum_{k=0}^{\infty} \frac{k}{\sqrt{k^{2}+1}}$$

Give an argument, similar to that given in the text for the harmonic series, to show that \(\sum_{k=1}^{\infty} \frac{1}{\sqrt{k}}\) diverges.
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