Can we say the sequence is {x-1/k2} for integers>0 ? I'm still a little confused.
I know that the interval (x-1/k, x+1/k) will be contained in a neighborhood of x, and thus has infinitely many points. I'm not sure why we are using 1/k though. I guess we are in essence using the definition of...
Homework Statement
Suppose x is an accumulation point of {an: n is a member of integers}. Show there is a subsequence of (an) that converges to x.
The Attempt at a Solution
I'm a little stuck on this one. I know that since x is an accumulation point then every neighborhood around x...
So, I might have got it...
I tried it by contradiction.
Assume x is not a member of S and assume it is not an accumulation point of S. If there is a neighborhood (x - epsilon, x + epsilon) containing x that does not have a point of S, then (x - epsilon) is an upper bound of S that's less than...
Homework Statement
Let S be a nonempty set of real numbers bounded from above and let x=supS. Prove x either belongs to the set or is an accumulation point of S.
Homework Equations
x is an accumulation point of S iff each neighborhood of x contains a member of S different from x. That...
Chose e>0. Let N be any positive integer greater than 1/e. Then, for n>=N we have
|1/(2n)-0| < |1/n|= (1/n) <= (1/N) < (1/(1/e)) = e
Thus, the sequence converges to 0??
Does that look right?
Homework Statement
Using the definition of convergence to prove that the sequence {2^(-n)} converges
Homework Equations
The Attempt at a Solution
So, I just don't think I am thinking straight or something. Here is what I got so far:
Chose e>0. Let N be any positive integer...
Homework Statement
Let x and y be real numbers with x<y and write an inequality involving a rational
number p/q capturing what we need to prove. Multiply everything in your inequality by q,
then explain why this means you want q to be large enough so that q(y-x)>1 . Explain
how you...
Homework Statement
Consider the following initial value problem:
y''+4y = 9t, 0<=t<2
...0, t>=2
Find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)
Homework Equations
The Attempt at a Solution
I am just having trouble with the step...
Homework Statement
Use the Laplace transform to solve the following initial value problem:
x' = 7 x + 5 y, y'= -2 x + e5t, x(0)=0, y(0)=0
Find the expressions you obtain by taking the Laplace transform of both differential equations and solving for Y(s) and X(s)
Homework Equations...
Homework Statement
Just a quick question concerning a Laplace transformation...
Find the Laplace transform of the following function:
f(t)=10t3/2-e(-7t)
Homework Equations
The Attempt at a Solution
I wasn't sure what to do with the t3/2 so I just followed the formula for t1/2...
Homework Statement
Solve the system.
dx/dt=[1 -4; 4 -7]*x with x(0)=[3; 2]Homework Equations
The Attempt at a Solution
I am apparently not getting this at all. Can someone walk me through it? I konw I have to first find the eigenvalues and eigenvectors:
(1-λ)(-7-λ)+16=0
λ2+6λ+9=0
λ=-3,-3
So...
Homework Statement
solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9]
Homework Equations
The Attempt at a Solution
I know I need to find the Eigenvalues but then I get a little confused from there.
(λ-3)(λ+3)=0
λ=3, -3