2 Hints

In this section, you’ll find hints for the current week’s problem sheet. Try and have a go without them first, but hopefully these will help you solve the problems.

1. This question is all about sequences!
   1. This one requires you to use the definition, so you can’t use any of the tests!
      1. Remember, if you make the denominator of a fraction smaller, you make the overall fraction larger.
      2. \(\sqrt[4]{n} = \sqrt{\sqrt{n}}\).
   2. Think about which tests you can apply for convergence. As a further hint, one of these sequences converges, and one doesn’t.
2. This question is all about series! Think about all the tests you’ve seen for series convergence, and also, think about some of the series you calculated last semester.
3. This questions is all about sets and bounds!
   1. Before you start calculating the \(\sup\),\(\inf\), etc., draw a graph — it’ll help you rewrite this set.
   2. Try writing out the first few components of the infinite union, this might suggest whether the set is bounded or not.