1. Homework Statement
Let s be the sum of the alternating series \sum(from n=1 to \infty)(-1)n+1an with n-th partial sum sn. Show that |s - sn| \leqan+1
2. Homework Equations
I know about Cauchy sequences, the Ratio test, the Root test
3. The Attempt at a Solution
I really have...
1. Homework Statement
Suppose that, on average, 70% of graduating students want 2 guest tickets for a graduation ceremony, 20% want 1 guest ticket and the remaining10% don't want any guest tickets.
(a) Let X be the number of tickets required by a randomly chosen student. Find the mean and...
1. Homework Statement
Suppose that f: R -> R is continuous on R and that lim (x -> \infty+)(f(x) = 0) and lim (x -> \infty-)(f(x)=0).
Prove that f is bounded on R
2. Homework Equations
I have got the proof of when f is continuous on [a,b] then f is bounded on[a,b] but I'm unsure as to...
1. Homework Statement
Let V = {differentiable f:R -> R}, a vector space over R. Take g1,g2,g3 in V where g1(x) = e^{}x, g2(x) = e^{}2x and g3(x) = e^{}3x.
Show that g1, g2 and g3 are distinct.
2. Homework Equations
If g1-g3 are linearly independent, it means that for any constant, k in F...
1. Homework Statement
Prove that lim n \rightarrow\infty 2^{}n/n! = 0
2. Homework Equations
This implies that 2^{}n/n! is a null sequence and so therefore this must hold:
(\forall E >0)(\existsN E N^{}+)(\foralln E N^{}+)[(n > N) \Rightarrow (|a_{}n| < E)
3. The Attempt at a...
1. Homework Statement
Let X be a random vairable which can only take three values: -1,0,1 and they each have the same probability. Let Y also be a random vairable defined by Y = X2. Show that
i) X and Y are not independent
ii) X and Y are uncorrelated
2. Homework Equations
To show...
1. Homework Statement
Solve: (2t+x) dx/dt + t = 0
2. Homework Equations
y' +p(X)y = q(x)
and y(x) = (\intu(x)q(x) + c)/u(x)
where u(x) = e\intp(x)dx
Note this u(x) is 2 to the power of the integral of p(x)
3. The Attempt at a Solution
(2t+x) dx/dt + t = 0 becomes:
dx/dt + t/(2t+x) = 0 by...
1. Homework Statement
Prove that (\foralln in the set of Natural numbers )[(n \geq 9) \Rightarrow (2n > 4n2 + 1)]
2. Homework Equations
To do proof by induction you must first prove for n = 1, then assume true for n and then show for n+1
3. The Attempt at a Solution
So for n=1 i have...
1. Homework Statement
A bag contains four dice labelled 1,...,4. The die labelled j has j white faces and (6-j) black faces, j = 1,...,4. A die is chosen at random from the bag and rolled. We define X = the number labelling the chosen die.
Y = {0 if the face showing on the die is black; 1 if...
1. Homework Statement
In this statement find the integer y such that x == y mod(n) where 0 =< y < n
x=2^71 n=23
2. Homework Equations
Let p be a prime number and let a be an integer which is not divisible by p. Then a^(p-1) == 1mod(p)
3. The Attempt at a Solution
I am really struggling...
Let f: X → Y and A is a subset of Y and B is a subset of Y. Prove that:
a) f⁻¹(A union B) = f⁻¹(A) union f⁻¹(B)
b) f⁻¹(A intersetion B) = f⁻¹(A) intersection f⁻¹(B).
I know that f⁻¹(A) = {x ε X : f(x) ε A}
and so f⁻¹(B) {x ε X : f(x) ε B}
but after that I really don't understand how to prove...