n! kn-k) (2) = 2t(rn, 2)! =n2( ) í)(n 2)(n(n 3)3):(2)2) i n(n-1)(n-2)(n-3) (2) 1 n(n-1)...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a) the distance di from Zo to Hi-{x E Rn : XTXǐ (b) the distance sk from ro to n1Hi, 1 <k< n (c) the distance mk from a'0 to ngk+1H,, 1-K n (d) calculate sk + mk 0) Problem 3: Let xi be given n mutually orthogonal vectors in Rn, and 20 є Rn be also given. Find: (a)...
1) Give a combinatorial proof of the following identity (0 <k<n): n2 k ---- = n.29-1 ke=0
How is the last step done done N-1 3 i⅔n2 e 10 10 N _ n-0 N-1 3 i⅔n2 e 10 10 N _ n-0
(5 points) Suppose the joint probability mass function (pmf) of integer- Y ī PlX = í,ys j) = (i + 2j)o, for 0 í valued random variables X and < 2,0 < j < 2, and i +j < 3, where c is a constant. In other words, the joint pmf of X and Y can be represented by the table: Y=2 |Y=0 Y=1 X=0| 0 2c 4c 3c 4c 5c X=21 2c (a) Find the constant c. (b) Compute...
show that if ch[k-n], h[k] > = Ži h*[kn] h[k] = Str], then I Herita, 1 = 1 K:-00 - ICWCTI
Kila K2 B B Tag K2 Kila i N2 N₂ A 0 K KIB KIB N K N species 1 species 2 (b) (a) Kila K2 B To 7 N₂ ž K2 Kila D E E X с A ܘܠ A 0 Ky K2/B 0 KB K N N (c) (d) In graph (d), in the lower right-hand region (point C), the combined dynamics the equilibrium point and the carrying capacity of species 1. move away from; toward O support;...
(1) Using the identity: n n! (2) want k k!(n - k)! for n > 1, prove the following identity: ()-20) + n2
Q-6e: Determine the big-O expression for the following T(N) function: T(1) = 1 T(N) = 2T(N – 1)+1 O 0(1) O O(log N) OO(N2) O O(N log N) O 0(2) OO(N)
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...