Reference no: EM131081268
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
![1013_Figure.png](https://secure.expertsmind.com/CMSImages/1013_Figure.png)
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 (sn) 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
![1151_Figure1.png](https://secure.expertsmind.com/CMSImages/1151_Figure1.png)
where α is a real constant.
(a) Prove that fn → f point-wise.
(b) Prove that fn → f uniformly if and only if limx→0^+ f(x) = α.