Sequences and series convergence – university Year 1 maths test to download as PDF

Let \(a_n=\dfrac{2n-1}{n+1}\).

1. Find the limit \(L\) and show that \(|a_n-L|=\dfrac3{n+1}\). (2 marks)
2. Prove from the definition that \(a_n\to L\). (2 marks)

1. Find \(\displaystyle\sum_{n=1}^\infty\dfrac4{(2n-1)(2n+1)}\). (2 marks)
2. Find \(\displaystyle\sum_{n=0}^\infty3\left(-\tfrac12\right)^n\). (2 marks)

1. Does \(\sum\dfrac n{n^3+1}\) converge? (2 marks)
2. Does \(\sum\dfrac{2^n}{n^2}\) converge? (2 marks)

Consider \(\displaystyle\sum_{n\ge1}\dfrac{(-1)^{n+1}}{n^2}\).

1. Show that it converges absolutely. (2 marks)
2. Estimate its sum with four terms and give a bound for the error. (2 marks)

Consider \(\displaystyle f(x)=\sum_{n\ge1}\dfrac{(x-1)^n}{n\,2^n}\).

1. Find the radius of convergence. (2 marks)
2. Find the interval of convergence, testing the endpoints. (2 marks)