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# Lecture 1: Exercises

### Suggestions for further reading

- Complex arithmetic: Lyons (chap. 4), Arfken (chap. 6.1)
- Infinite series: Arfken (chap. 5.1)

### Questions for Review

- Do all quadratic equations have a real solution? Give examples!
- What is the imaginary unit?
- What is a complex number?
- How can complex numbers be represented graphically?
- How does one calculate the absolute value and the argument of a complex
number?
- How does one add complex numbers? How can this addition be represented
graphically?
- What is the triangle inequality?
- How does one multiply complex numbers? What are the absolute value
and the argument of a product of complex numbers?
- What is the inverse of a complex number? What are its absolute value
and its argument?
- How is the complex conjugate of a complex number defined? What does
this correspond to graphically?
- What is the complex conjugate of a sum/product of complex numbers?
- What is an infinite sequence? How can an infinite sequence be represented?
- What does it mean to say that an infinite sequence converges?
- What is a monotonic sequence? Under which conditions does it converge?
- What is an infinite series? Give examples!
- What does it mean to say that an infinite series converges?
- What is the geometric series? Does it converge? Why?
- What is the harmonic series? Does it converge? Why?

### Problems