
translate from the question.
1.suppose xn
2.prove that Xn rises and is limited above 2 ^ n-1 ≤ n!
3.use the fact that
4.for each
Now since xn
is bounded and monotonically incresing.
aso xn will converge also .
translate from the question. 1.suppose xn 2.prove that Xn rises and is limited above 2 ^...
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
2) Prove that 1 + 3n < 4n for all n > 1. /5 Marks/
Prove that if |A| = |Band [B<|A|, then |A| = |B).
(1) Let {fn} < C[a, b], and let {xn} c [a, b]. Suppose that fn + f uniformly on [a, b] and In + x (as n +00. Show that limn7 fn(2n) = f(x). [3] To a + + ( f or intuico te fan a ant [Dannt u
Prove the ratio test . What does this tell you if
exists?
(Ratio test) If
for all sufficiently large n and some
r < 1, then
converges absolutely; while if
for
all sufficiently large n, then
diverges.
lim |.1n+1/01 700 In+1/xn < We were unable to transcribe this image2x+1/2 > 1 We were unable to transcribe this image
let a,b > 0 . Prove that
DI < Val
Let X be a continuous random variable. Prove that: P(21-; < X < xạ) = 1 - a.
Let ne Nj. Prove that n < 2(6(n)).
(2) Prove by induction that for all integers n > 2. Hint: 2n-1-2n2,
Given f(x,y) = 2 ; 0 <X<y< 1 a. Prove that f(x,y) is a joint pdf b. Find the correlation coefficient of X and Y