5.2 Computing Limits of Sequences
5.2.1 Limits of Sequences and Functions
- If \(f(x)\) is a function and \(a_n\) is a sequence such that \(f(n)=a_n\) for sufficiently large integers \(n\), then \(\lim_{x\to\infty}f(x)=L\) implies \(\lim_{n\to\infty}a_n=L\).
- Therefore all the rules for evaluating \(\lim_{x\to\infty}f(x)\) extend to evaluating \(\lim_{n\to\infty}a_n\).
- Example Use factoring to compute \(\lim_{n\to\infty}\frac{4+n}{n^3+1}\).
- Example Use L’Hopital’s Rule to prove that any sequence defined by the formula \(a_n=\frac{n^2+3}{4-5n^2}\) converges to \(-\frac{1}{5}\).
- Example Use the squeeze theorem to compute \(\lim_{n\to\infty}\frac{\sin n}{n}\).
5.2.2 Common Limits
- The following limits are often useful:
- \(\lim_{n\to\infty} x = x\)
- \(\lim_{n\to\infty} \frac{1}{n} = 0\)
- \(\lim_{n\to\infty} \frac{\ln n}{n} = 0\)
- \(\lim_{n\to\infty} \sqrt[n]{p(n)} = 1\) where \(p(n)\) is a polynomial
- \(\lim_{n\to\infty} x^n = 0\), \(|x|<1\)
- \(\lim_{n\to\infty} (1+\frac{x}{n})^n=e^x\)
- \(\lim_{n\to\infty} \frac{x^n}{n!}=0\)
- Example Find \(\lim_{n\to\infty}\frac{\ln(n^3)}{n}\).
- Example Find \(\lim_{n\to\infty}\frac{3^n+1}{n!}\).
- Example Find \(\lim_{n\to\infty}(4n)^{1/n}\).
5.2.3 Monotonic and Bounded Sequences
- A sequence \(\<a_n\>_{n=i}^\infty\) is bounded if there exist real numbers \(A,B\) such that \(A\leq a_n\leq B\) for all integers \(n\geq i\).
- Example Is the sequence \(\<a_n\>_{n=1}^\infty\) where \(a_n=\frac{n+1}{n}\) bounded?
- Example Is the sequence \(\<b_n\>_{n=0}^\infty\) given by \(b_n=\frac{n}{(-3)^n}\) bounded?
- A sequence is monotonic if it either never increases or never decreases.
- Example Is the sequence \(\<a_n\>_{n=1}^\infty\) where \(a_n=\frac{n+1}{n}\) monotonic?
- Example Is the sequence \(\<b_n\>_{n=0}^\infty\) given by \(b_n=\frac{n}{(-3)^n}\) monotonic?
- The Monotonic Sequence Theorem states that all bounded monotonic sequences converge.
Exercises for 5.2
- Use factoring to compute \(\displaystyle\lim_{n\to\infty}\frac{n-4n^2}{2n^2+7}\).
- Use L’Hopital’s Rule to prove that \(\displaystyle\frac{\ln n}{n}\to 0\).
- Use the squeeze theorem to compute \(\displaystyle\lim_{n\to\infty}\frac{\cos n}{n\ln n}\).
- Find \(\displaystyle\lim_{n\to\infty}\frac{\sin n + 3n^2}{n^2+1}\).
- Find \(\displaystyle\lim_{n\to\infty}\frac{\ln(n^n)}{n^2}\).
- Find \(\displaystyle\lim_{n\to\infty}(5n^3)^{2/n}\).
- Find \(\displaystyle\lim_{n\to\infty}(\frac{1}{\pi})^{3n}\).
- Find \(\displaystyle\lim_{n\to\infty}(\frac{1}{2}+\frac{1}{n})^n\).
- Find \(\displaystyle\lim_{n\to\infty} \frac{\frac{(n+2)!}{2^n}}{\frac{3n^2n!}{2^{n+1}}}\).
- Based on its first few terms, does the sequence \(\<a_n\>_{n=2}^\infty\) where \(a_n=\frac{2+n^2}{n^2-1}\) appear bounded? Monotonic? Does it appear to converge?
- Based on its first few terms, does the sequence \(\<b_n\>_{n=0}^\infty\) where \(b_n=(-3)^n\) appear bounded? Monotonic? Does it appear to converge?
- Based on its first few terms, does the sequence \(\<y_n\>_{n=1}^\infty\) where \(y_n=(-\frac{1}{2})^n\) appear bounded? Monotonic? Does it appear to converge?
- Prove that \(\displaystyle\lim_{n\to\infty} (1+\frac{x}{n})^n=e^x\) by considering the function version \(\displaystyle L=\lim_{t\to\infty} (1+\frac{x}{t})^t\) and taking the natural log of both sides of the equality. Use L’Hopital to solve this limit, showing that \(\ln L=x\) and therefore \(L=e^x\).
- Find \(\lim_{n\to\infty}\frac{n!\cos n}{(n+1)!}\).
- Find \(\lim_{n\to\infty}\frac{(3+n)^n}{n^n}\).
- Which of these statements seems most appropriate for describing the
sequence whose initial terms are
\(\<\frac{1}{4},-\frac{1}{6},\frac{1}{8},-\frac{1}{10},
\frac{1}{12},\dots\>\)?
- The sequence is bounded and monotonic, so it converges by the Monotonic Sequence Theorem.
- The sequence is not monotonic and not bounded, so it diverges by the Monotonic Sequence Theorem.
- The sequence is bounded, but not monotonic, so the Monotonic Sequence Theorem doesn’t apply. However, it does appear to converge to \(0\) anyway.
Textbook References
- University Calculus: Early Transcendentals (3rd Ed)
- 9.1