MAS1006 Advanced Calculus

Test: Wednesday 25th April 2001, 12.05–12.55 pm

Please attempt all questions

1. (a) Find the general solution of the following differential equation:

\frac{dy}{dx} = \frac{y − 2}{x + 1}.

Also find the particular solution satisfying y(1) = 0. (10)

(b) Find the general solution of the following differential equation:

y′′ − 2y′ + 2y = 2x − 1.

(10)

2. Determine, giving reasons, whether each of the following sequences converges:

(a) \frac{n^{2} − n sin n}{n^{2}},  (b) \frac{(1 − n^{2})^{3}}{(1 + n^{3})^{2}} ,  (c) \frac{(−1)^{n}n^{2}}{n^{2} + 1} .

Find the limit of each convergent sequence. (10,10,10)

3. Determine, stating clearly which convergence test you are using, whether each of the following series converges:

(a) ∑_{n=1}^{∞} \frac{(−1)^{n}}{\sqrt{n}} ,  (b) ∑_{n=1}^{∞} \frac{\sqrt{n}}{2n^{2} − 1},  (c) ∑_{n=2}^{∞} \frac{1}{ln n}.

[You may use, without proof, the criterion for when ∑_{n=1}^{∞} 1/n^{k} is convergent.] (10,10,10)

4. Find the radius of convergence of each of the following power series:

(a) ∑_{n=1}^{∞} \frac{n(−2x)^{n}}{n^{2} + 1} ,  (b) ∑_{n=1}^{∞} \frac{n!^{2}x^{n}}{(2n)!} .  (10,10)

[Total marks: 100]

Examiner: Robin Chapman