Prove that for all n ≥ 1, nn ≥ n!.
for n=1: --------- LHS = 1^1 = 1 RHS = 1! = 1 so, LHS >= RHS for n=1 let's assume LHS >= RHS for n=k: --------------------------------- k^k >= k! for n=k+1: ----------- LHS = (k+1)^(k+1) = (k+1)^k * (k+1) >= k^k * (k+1) >= k! * (k+1) >= (k+1)! >= RHS so, LHS >= RHS for n=k+1, given that LHS >= RHS for n=k. hence proved by induction.
for n=1:
LHS = 1^1 = 1,RHS = 1! = 1
so, LHS >= RHS for n=1
let's assume LHS >= RHS for n=k:
k^k >= k!
for n=k+1:
LHS = (k+1)^(k+1) = (k+1)^k * (k+1) >= k^k * (k+1) >= k! * (k+1) >= (k+1)! >= RHS
so, LHS >= RHS for n=k+1
12. Prove that if n >m then the number of m-cycles in Sis given by nn-1)(n-2)... (n-m+1)
state and prove divergence or convergence for each of the following
series.
f. 5 nn |(n+3) n= ln 00 g. (2n-1) (n!) n=1 h. į vn + cos n n n=1 i. 2"n? n!
Let AA be an n×nn×n matrix. Prove that if x⃗ x→ is an eigenvector of AA corresponding to the eigenvalue λλ, then x⃗ x→ is also an eigenvector of A+cIA+cI, where cc is a scalar. Moreover, find the corresponding eigenvalue of A+cIA+cI.
4. (20 points) Prove P(Ln MnN) PLIM nN)P(MN)P(N).
Please Prove your answers mathematically, I need clear
writing
PROBLEM2 y(n) = 2 x(n-2) sin (nn) Determine whether the system Justify your answer and 1. linear 2. time invariant 3. Stahle
all three questions please. thank you
Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....
Prove that Şi = n(n+1) for all integers n 2 1.
help please and thank you
5. Prove that --> 2(n+1 - 1) for all n e Zt. 6. Prove that n < 2" for all n e Z.
Prove by Induction
24.) Prove that for all natural numbers n 2 5, (n+1)! 2n+3 b.) Prove that for all integers n (Hint: First prove the following lemma: If n E Z, n2 6 then then proceed with your proof.
Problem 30. Prove that N, Z, Q and R are infinite sets. (HINT: Prove by induction on n that is f: NN then (3k N(Vj Nn)k> f(j). Then conclude that f cannot possibly be onto N. A similar strategy works for Z, gq and R as well.)