2 Hints

As per usual, here’s where you’ll find the problem sheet hints!

  1. Use the definition! Try and follow a similar format to what we did in tutorials. Make sure to write things logically, and ensure that you’ve satisfied each part of the definition.
  2. Again, this is an exercise in using the definition of convergence.
  3. Without loss of generality, assume that \(\lvert z_1 \rvert \geq \lvert z_2 \rvert.\) The advice here is that if you ever get stuck in Analysis, either add \(0\), or multiply by \(1\) in a clever way. Try and do one of these tricks to allow you to apply the triangle inequality.
    1. This is an if and only if statement, so there are two things to prove! Try and manipulate the definition of convergence for one side of the \(\Leftrightarrow\) statement to resemble the other.
    2. Recall what it means for a complex number to be in modulus-argument form.
    3. Definition again, and follow the hint on the sheet.
    4. I’ll leave this to you! Think simple!
  5. For a) and b), De Moivre’s Theorem will come in handy! For c), when bounding, remember that if you make a denominator smaller, the whole fraction will get bigger.