New! Section 2.2 – Sequences
Instructions
- The first videos below were recorded specifically for MATH 147. Exercises 3 and later are similar problems from MATH 152.
- The first videos below explain the concepts in this section.
- This page also includes exercises that you should attempt to solve yourself. You can check your answers and watch the videos explaining how to solve the exercises.
Concepts
- An introduction to sequences and their limits along with the limit of recurrence equations
- Finding the first few terms and limit of a sequence
- Finding the first few terms and limit of a recursive sequence
Similar Material From Other Courses
The following are links and videos from other courses that cover this same material. This might be useful, but please note that other courses might cover the material differently.Section 11.1 – Sequences
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Solve the following:
- Write the first five terms, \(a_0,\) \(a_1,\) \(a_2,\) \(a_3,\) and \(a_4,\) of the sequence: \[a_n=\dfrac{(-1)^{n+1}(n+3)}{4^n}\]
- Compute \(\displaystyle \lim_{n\rightarrow \infty} a_n,\) if it exists.
- \(\left\{ -3, 1,-\dfrac{5}{16},\dfrac{6}{64}, -\dfrac{7}{256}, \ldots \right\}\)
- \(\displaystyle \lim_{n\rightarrow\infty} a_n = 0\)
2. Solve the following:
- Write the first five terms, \(a_0,\) \(a_1,\) \(a_2,\) \(a_3,\) and \(a_4,\) of the sequence: \[a_{n+1}=\dfrac{1}{5}a_n-(a_n)^2 \quad \textrm{ and } \quad a_0=2\]
- Compute all fixed points of \(\{a_n\},\) then guess which fixed point is the limiting value for the given initial condition.
- \(a_0= 2,\) \(a_1=-3.6,\) \(a_2=-13.68,\) \(a_3=-189.8784,\) \(a_4=-36091.78247,\) \(a_5\approx -1302623980\)
- Fixed points: \(a=0\) or \(a=-\dfrac{4}{5}\)
3. Find a formula for the general term, \(a_n\), of the sequence assuming that the pattern of the first few terms continues. Give the formula so that the first term is \(a_1\).
\[\left\{\frac{5}{27}, -\frac{9}{64}, \frac{13}{125}, -\frac{17}{216}, \ldots\right\} \]
\(a_n=\dfrac{ (-1)^{n+1}(4n+1)}{(n+2)^3}\)
4. Determine whether each sequence converges or diverges. If it converges, find the limit. \[a_n=\dfrac{\ln(\ln(2n))}{5n}\]
Converges to 0
5. Determine whether each sequence converges or diverges. If it converges, find the limit. \[a_n=\frac{(-1)^n n^3}{7n^3-1}\]
Diverges
6. Determine whether each sequence converges or diverges. If it converges, find the limit. \[a_n=\arctan\left(\dfrac{1-n^2}{3n+1}\right)\]
Converges to \(-\dfrac{\pi}{2}\)
7. Determine whether the following sequence is increasing, decreasing, or not monotonic. Also, determine if the sequence is bounded. \[a_n=\ln(8+9n)-\ln(3n-2)\]
Bounded and Decreasing
8. Determine whether the following sequence is increasing, decreasing, or not monotonic. Also, determine if the sequence is bounded. \[a_n=\tan\left( \dfrac{(2n-1)\pi}{4}\right)\]
Video errata: The speaker forgot to say that the sequence is not monotonic since the terms are alternating between \(-1\) and \(1.\)
Bounded and Not Monotonic
9. The sequence below is bounded and increasing (check!). Does it converge and if so, what to? \[a_1=3 \qquad \qquad a_{n+1}=\dfrac{5a_n-4}{a_n}\]
It converges to 4.