Math 104 midterm 2 show that the series diverges at x 12

Question

Math 104: Midterm 2-

1. Consider the power series

_n=1∑^∞(2x)^{n^2}/n.

(a) Show that the series diverges at x = 1/2 and converges at x = -1/2.

(b) What is the radius of convergence of the series? Either calculate it explicitly, or justify carefully using part (a).

2. Consider the function       

defined for all x, y ∈ R.

(a) Prove that d is a metric on R.

(b) Find the interior of [0, 1] with respect to d.

(c) Suppose that (s_n) is a Cauchy sequence in R with respect to d. Prove that it is a convergent sequence with respect to d.

3. Consider the functions f(x) = x² (2 - x) and g(x) = |f(x)| defined for all x ∈ R.

(a) Sketch f and g over the domain -1 ≤ x ≤ 3.

(b) Use the ε-δ property to prove that g is continuous at x = 2.

(c) Prove that there are at least four solutions to the equation g(x) = 1/2.

4. Let f be a real-valued function on (0, 1). Define a sequence of functions as

where α is a real constant.

(a) Prove that f_n → f point-wise.

(b) Prove that f_n → f uniformly if and only if lim_x→0^+ f(x) = α.