1.1.7 For what values of p and q will Σχ2 nrin n-converge? p< I, all q...
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
Prove using mathematical induction that 3" + 4" < 5" for all n > 2.
For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
Evaluate the piecewise defined function at the indicated values (x2 f(x) if x -1 6x if 1 < x s 1 = -1 if x > 1 f(-3) (- 3 2 f(-1) f(0) = f(30) =
Find the minimum and the maximum values of |z2 + (p + 1)i| on the closed disc {z ∈ C : |z| ≤ q + 1} Find the minimum and the maximum values of 122 + (p + 1)i| on the closed disc {z € C: |Z| <q+1}.
Problem 5.1.3. Prove by induction on n that (1+ n < n for every integer n > 3.
4. Consider our standard LP: maxc.x subject to Ax <b and x > 0. Assume every entry of A is strictly positive and b > 0. Deduce that the LP has an optimal solution.
#36. max sa, Q..., an? <3 show that Itt tunta-an> 3. (atacteuta - + - az 3-a,
5. (10 points) Let p="x < y", q="x < 1", and r="y > 0". Using ~, 1, V write the following statements in terms of the symbols p, q, and r. (a) 0 <y < x < 1. (b) 1 < x <y<0.
(5) Use induction to show that Ig(n) <n for all n > 1.