MATH 395 PROBLEMS 1 - REVIEW PROBLEMS

IGOR KRIZ

Regular problems:

1. Suppose a matrix A = ( \begin{matrix}a & b \\ c & d \end{matrix} ) has an inverse A^{−1} (i.e. a matrix such that

AA^{−1} = A^{−1}A = I). Find an explicit formula for A^{−1}.

2. Let S = ∫_{0}^{1} e^{t}/(t + 1)dt. Express the values of the following integrals in terms of S:

(a) ∫_{a−1}^{a} \frac{e^{−t}}{t−a−1}dt.

(b) ∫_{0}^{1} e^{t}ln(1 + t)dt

3. Let f(r, t) = t^{n}e^{−r^{2}/(4t)}. Find a value of the constant n such that f satisfies the following equation:

\frac{∂f}{∂t} = \frac{1}{r^{2}} \frac{∂}{∂r} (r^{2} \frac{∂f}{∂r} ) .

4. Evaluate the following infinite sums:

(a) ∑_{n=0}^{∞} \frac{1}{2^{n}}.

(b) ∑_{n=0}^{∞} \frac{(−1)^{n}2^{2n}π^{2n}}{(2n)!} .

(c) ∑_{n=0}^{∞} \frac{1}{3^{n}(n+1)}.

Challenge problems:

5. (a) Prove that the series

∑_{n=0}^{∞} 2^{n} sin \frac{1}{3^{n}x}

converges uniformly on [a, ∞) for a > 0. [Hint: lim_{h→0}(sin h)/h = 1.]

(b) By considering the sum from N to ∞ for x = 2/(π3^{N}), show that the series does not converge uniformly on (0, ∞).

6. Find the set of all points ( \begin{matrix}a \\ b \\ c \end{matrix} ) ∈ ℝ^{3} for which the two spheres (x − a)^{2} +

(y − b)^{2} + (z − c)^{2} = 1 and x^{2} + y^{2} + z^{2} = 1 intersect orthogonally. (Their tangent planes should be perpendicular at each point of intersection.)