3 Hints

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

  1. Try applying one of the tests for monotonicity. For the limit, if you need to use any theorems anywhere, state them!
  2. See the hint for question 1 above.
  3. Use the definitions of monotone increasing and unbounded together to deduce the sequence satisfies the definition of \(a_n \to \infty\).
  4. If \(a_n \neq 0 \, \; \forall n \in \mathbb{N}\), then this would just be an application of the algebra of (infinite) limits! Since you don’t know if this is the case, you’ll need to use the definition again for this question. You’ll end up with an inequality of the form \(a_n^2 > g(\epsilon)\), where \(g\) is a rational function of \(\epsilon\), at some point in your solution. Take cases on the sign of the numerator of \(g(\epsilon)\) to find the required \(N\) in the definition of limit.
  5. This is similar to Tutorial Question 3 off Exercise Sheet 6. Try a few terms of the sequence to get a feel for what’s happening first. Note that you’re not explicitly told to find the limit, but it’s really worth doing if you can!
    1. Try applying one of the tests for monotonicity.
    2. In the inductive step, you’ll encounter two fractions. Get them over a common denominator, and then ‘kill off’ some terms in the numerator (briefly justifying why you can do this). Also, make sure to format your induction proof logically!
    3. Seen any good theorems about bounded monotonic sequences lately?4

  1. As on the sheet, the limit of the sequence \((a_n)\) in this question is \(\pi^2/6.\) To establish this, we have to study Fourier series. It turns out that for \(f:[-\pi,\pi]\to\mathbb{R}\) given by \(f(x) = x^2,\) we can write \(f\) as a sum of cosines, namely \[f(x) = \frac{\pi^2}{3} + 4\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}\cos(nx).\] Evaluating this series at \(x=\pi\) (and assuming the infinite series behaves nicely), we find that \[\pi^2 = \frac{\pi^2}{3} + 4\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}\cdot(-1)^n,\] from which \[\sum_{n=1}^{\infty}\frac{1}{n^2} = \lim_{n\to \infty}a_n = \frac{\pi^2}{6}.\] For more information, take Vector Calculus and Partial Differential Equations (MA20223) in Year 2!↩︎