UNIVERSITY OF EXETER DEPARTMENT OF MATHEMATICS Mathematical Methods 2: CA Test 6th March 1997, 11.05-11.55am Solutions should be handed in for all questions

1. Test for convergence the following series, clearly stating in each case which

convergence test you are using: (a)

∑_{n=1}^{∞} \frac{3n^{2} + 1}{n3^{n}}

(b)

∑_{n=1}^{∞} \frac{1}{3n + 2}

(c)

∑_{n=1}^{∞} \frac{1}{5n^{\sqrt{2}}}

[4 marks each]

2. Find the radius of convergence of the following series:

∑_{n=0}^{∞} \frac{(−1)^{n}3^{n}x^{n}}{n^{2}}

[8 marks]

3. Evaluate the limit of the following expression:

lim_{x→π/2} (\frac{ln(1 + cos x)}{ln(1 + sin 2x)})

[8 marks]

4. (a) Let z = f(x, y), where x = r cos θ and y = r sin θ. Find \frac{∂^{2}z}{∂θ^{2}} in terms of partial

derivatives with respect to x and y of first and second orders. [8 marks]

(b) Let z = f(x, v, t) where x = x(v, t). Use differentials to find an expression for

\frac{∂z}{∂t}. [6 marks]

5. Show that the function w = sin(x + ct) satisfies the equation

\frac{∂^{2}w}{∂t^{2}} = c^{2} \frac{∂^{2}w}{∂x^{2}}

[8 marks]

MathsMethsTest 6.03.97