| Prove that for n e N, n > 0, we have 1 x 1!+ 2 x 2!+... tnx n! = (n + 1)! - 1.
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
- Let fm (x)= 7* (0<x< 1). Show that { {m} -, converges pointwise on [0, 1]. If f(x)= lim fn(x) (0<x< 1), is there an N EI such that In(x)-f(x)}< (n>N) for all x € [0, 1] simultaneously?
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.
IDY in < oo and lim - Yn < 0o. Prove that lim,+ 1. Let In > 0. Yn > 0 such that lim,- Yn) < lim,-- In lim,+ Yn: i tn < oo and lim yn < . Prove that lim. In 1. Let In 20, yn 0 such that lim Yn) < limn+In lim + Yr
Prove that A = B for: A = {(x,y) e Rº : +y/<1} B = {(z,y) € RP: (71+ y)² < 1}
5. Given the probability density f(x)= for -0<x<00, find k. 1+ 2 Jor -
(6) Let A denote an m x n matrix. Prove that rank A < 1 if and only if A = BC. Where B is an m x 1 matrix and C is a 1 xn matrix. Solution (7) Do the following: (a) Use proof by induction to find a formula for for all positive integers n and for alld E R. Solution ... 2 for all positive (b) Find a closed formula for each entry of A" where A...
Let X and Y have join density 6 f(x, y) =-(x + y)2, 0 < x < 1, 0 < y < 1
5. (4 pts) Let X(ej) be the DTFT of a signal x[n] which is known to be zero for n < 0 and n > 3. We know X(eja) for four values of N as follows. X(@j0) = 10, X(eja/2) = 5 – 5j, X(ejt) = 0, X(ej37/2) = 5 + 5j (a) (3 pts) Find x[n]. (Hint: Compute the IDFT) (b) (1 pts) Find X(ej?).