Recent content by Hoblitz

Applying the Ratio Test, we have the sequence \frac{(n+1)n^n}{(n+1)^{(n+1)}} = \left(\frac{n}{n+1}\right)^n = \left(\frac{1}{1 + \frac{1}{n}}\right)^n Taking the limit as n goes to infinity... this might look like a familiar limit. Then recall that the Radius of convergence is the...

recall that for square matrices det(AB) = det(A)det(B) and for an invertible square matrix, det(B^{-1}) = det(B)^{-1} How can you apply these?

It may help to know that every convergent sequence in \mathBB{R} contains a monotone subsequence. The power function f(x) = x^a preserves monotonicity (although it may reverse it for negative powers). What would be a bound for such a sequence? Why would it have to be the least upper...

Well, look at f_k carefully: it has exactly two real roots no matter what k is: zero and 1. What is happening outside of the interval [0,1]? Inside the interval, it's a little more interesting. On the interval [0,1], for arbitrary k > 0 where does f_k achieve its maximum value, and what is...

I find it easier to think about these questions by picking arbitrary x_0 in the interval and seeing what happens. If you pick zero, f_k (0) = 0 and this is trivial. If x_0 is non-zero, indeed you can find large enough k such that \frac{1}{k} < x_0 , and for that k and larger, f_k (x_0) = 0...

Level curves f(x,y) = k are the sets { (x,y) | f(x,y) = k}. Now think geometrically: for a fixed k, what is the graph of k = (x + 1)^2 + y^2 ? (hint: rewrite k = (sqrt(k))^2 and this should start to look like a very familiar object). You could always try plotting a few points to get the gist of...

I can tell for the first one you're nearly there... after you've rewritten lim (x to 0) sin(1/x) / (1 / x) with u = 1/x, we have lim (u to infinity) sin (u) / u. In this case, how does the Squeeze theorem apply? For the second case, you might consider L' Hospital's rule, or the power series...

Indeed I can think a few unbounded functions f which are integrable on an interval [a,b], so I can see why my idea would certainly fail. I'll try looking at this from a new perspective and see what I get. What I understand to be happening is that the rapidly quickening oscillations of cosine are...

So the first equation is from the true distribution of x - y under H(a) which is Normal(10, 625/n) due to independence. The second equation comes form the upper tail test of x - y, which only rejects when the test statistic is greater than (in this case) 1.645. Seems to me you've done...

Homework Statement The problem is as follows: Let f be a real valued function that is Riemann integrable on [a,b]. Show that \lim_{\lambda \rightarrow \infty} \int_{a}^{b} f(x)\cos(\lambda x)dx = 0 . Homework Equations I am freely able to use the fact that the product of...