2 Hints

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

1. In this one, remember what we did in tutorials. One uses Leibniz, one can be done with Leibniz (but its quicker not to), and one diverges. Make sure you check the hypotheses of any test you use!
2. 1. Work out an expression for \(\frac{a_{n+1}}{a_n}\) (this will depend on whether \(n\) is odd or even).
   2. Check whether \((a_n)\) is an increasing or decreasing sequence first: this will help you calculate the \(\limsup\).
3. To begin here, you’ve seen a similar trick for the square roots in previous problem sheets. Think about your tests for convergence again on this one!
4. Use the alternative characterisation of suprema (Theorem 3.2 in the lecture notes).