2 Hints

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

1. Try using a similar argument to the one used in tutorial question 1 (i.e. use the fact that the sequence can be split into odd and even cases to your advantage)

2. For this question, think about what it means for a series to be convergent. You’ll also want to split the terms of the series up in some way. (Think of tutorial question 3.)

3. For the first part, think induction. The only other thing I’ll say is to make sure you state all the main results you use!

4. This is similar to homework question 2 and tutorial question 3.

5. Ok, this one seems a bit evil, but you’ve seen something similar in the proof of Lemma 4.1 from the lecture notes. To start with, there are two cases: \[\limsup_{n\to\infty}\frac{a_{n+1}}{a_n} = \infty, \quad \text{and} \quad \limsup_{n\to\infty}\frac{a_{n+1}}{a_n}<\infty.\] In the first case, there is nothing to prove (but you still need to tell me this in your answer!). For the second case, set \[\alpha = \limsup_{n\to\infty}\frac{a_{n+1}}{a_n}\] and consider an arbitrary \(\beta > \alpha.\) From this point, you need to do two things:

   • Explain why there exists \(N\in\mathbb{N}\) such that \[\frac{a_{n+1}}{a_n} \leq \beta \;\; \forall n \geq N.\]
   • Prove that for all \(n \geq N\), \[a_n \leq a_N\beta^{-N}\beta^n.\]

   Manipulate the result of this second bullet point, apply the results from Exercise Sheet 3, Tutorial Question 3 to conclude… and then go and grab a cup of tea or something.