Question 1 - Infinite Sequences.
(a). Determine an infinite sequence that satisfies the following .
. .
(i) An infinite sequence that is bounded below, decreasing, and
convergent
(ii) An infinite sequence that is bounded above and
divergent
(iii) An infinite sequence that is monotonic and converges to 1
as n → ∞
(iv) An infinite sequence that is neither increasing nor
decreasing and converges to 0 as n → ∞
(b). Given the recurrence relation an = an−1 +...
2. (a) Find the values of a and b such that the eigenvalues of A
= |a 1 are 2 and -5. (b) Find the values of a, b and c such that
the eigenvalues of A = | 0 1 0 | 0 0 1 | a b c are 3, -2, and
5.
b 1|
i)Show that infinite decidable language has infinite decidable
subset ?
ii)Show that any infinite decidable language L has an infinite
decidable subset J with the property that L − J is also infinite.
iii. Does the statement in part i of this problem still true if
L is only recognizable ? Show or Counter example.
No Spam please.
Show that the probability that all permutations of the sequence
1, 2, . . . , n have no number i being still in the ith position is
less than 0.37 if n is large enough. Show all your work.
Let (sn) be a sequence that converges.
(a) Show that if sn ≥ a for all but finitely many n,
then lim sn ≥ a.
(b) Show that if sn ≤ b for all but finitely many n,
then lim sn ≤ b.
(c) Conclude that if all but finitely many sn belong to [a,b],
then lim sn belongs to [a, b].
The Fibonacci sequence is an infinite sequence of numbers that
have important consequences for theoretical mathematics and
applications to arrangement of flower petals, population growth of
rabbits, and genetics. For each natural number n ≥ 1, the nth
Fibonacci number fn is defined inductively by
f1 = 1, f2 = 2, and fn+2 = fn+1 + fn
(a) Compute the first 8 Fibonacci numbers f1, · · · , f8.
(b) Show that for all natural numbers n, if α...
Find the eigenvalues and eigenfunctions of the given boundary
value problem. Assume that all eigenvalues are real. (Let
n represent an arbitrary positive number.)
y''+λy=
0,
y(0)= 0,
y'(π)= 0