For all n E N prove that 0 <e- > < 2 k!“ (n + 1)! k=0 Hint: Think about Taylor approximations of the function e".
.n= n(n-1)(n+1) for all n > 2. 12. Use induction to prove (1 : 2) +(2-3)+(3-4) +...+(n-1).n [9 points) 3
4. Let {Sn, n > 0} be a symmetric Random Walk on Z. with So-0. Defined Y max{Sk, 1 Sk n, for n 2 0, prove, thanks to a counterexample, that Y is not a Markov Chain.
Use induction to prove that 0–0 4j3 = n4 + 2n3 + n2 where n > 0.
7. Suppose X and Y have joint pdf f(x,y) = 24x y if x >0, y>0,x+y<1 and 0 otherwise. Find P(Y > 2x).
Problem 44) Prove: n!> 2" for n24. Problem 45) Prove by induction: For n>0·AT- i=1
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
Prove or Disprove that: If a > 0 and b are two rational numbers, then a' is a rational number.
Prove by induction that 1 1 15 15 + 1 35 35 + ... + = n 2n + 1 for every n > 1 4n2
Use induction and Pascal's identity to prove that (7) = 2" where n > 0.