Review for quiz 2
We have our first quiz in just a couple of days - on Wednesday, October 21st. The problems on the quiz will be a subset of variations of these problems.
Problems
Write down the following definitions
- Convergent series
Definition 4.1.12 of our text
- Absolute convergent series
Definition 4.1.13 of our text
- Convergent series
Find the largest open disks on which each of the following series converges:
- \(\displaystyle \sum_{n=1}^{\infty} \frac{n^2}{n!} z^n\)
- \(\displaystyle \sum_{n=1}^{\infty} \frac{n^3}{3^n} (z-(4+3i))^n\)
Note that degenerate cases, such as a single point or all of \(\mathbb{C}\) are possible.
- \(\displaystyle \sum_{n=1}^{\infty} \frac{n^2}{n!} z^n\)
Use the geometric series formula to evaluate the following series:
- \(\displaystyle \sum_{n=0}^{\infty} \frac{1}{(1+i)^n}\)
- \(\displaystyle \sum_{n=3}^{\infty} \frac{2^{n+2}}{(3+4i)^n}\)
- \(\displaystyle \sum_{n=0}^{\infty} \frac{1}{(1+i)^n}\)
Starting with the series representation \[ \arctan(z) = \sum_{n=0}^{\infty} (-1)^n \frac{z^{2 n+1}}{2 n+1}, \] find a power series representation of \[ \int \arctan(z^2) \, dz. \] What is the largest open disk on which your power series is valid?
Questions
Follow the “Reply on WriteTech” button to ask a question on or make a comment about this review sheet. Your questions and comments will auto-magically appear below. You can feel free to respond to other students questions and comments. I might chime in as well.