2 Hints

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

    1. At this stage of the course, you’ll have to use the definition of convergence. Apply this, and follow the hint on the sheet.
    2. This one you can apply AoL! However, you should explain why you can use AoL, namely: which convergence results from lectures are you applying?
  2. Again, this is one you have to use the definition on (for the time being, at least). Remember, if you make the denominator of a fraction smaller, you’ll make the overall fraction bigger.
  3. For the first bit, you can use AoL! For the second, you’ll have to use the definition, making a specific choice of \(\epsilon.\)
  4. This is very similar to the first question from tutorials, so here’s a few (vague-ish) hints:
    1. Try and find expressions to simplify the sums.
    2. Again, you’ll want to find an explicit expression for the sum of square numbers.
    3. Firstly, what do you get if you factorise \(a^3 - b^3\)? The result of Tutorial Question 2 will also be useful here.
    4. Evaluate \(\cos(2n\pi).\)
    5. Simplify the expression first.
    6. You’ll have to use the definition again here.
  5. This is another definition question. Use the knowledge that \((a_n)\) converges (i.e. what does this mean explicitly?) to conclude something about the convergence of \((b_n)\).