Recent content by Euler2718

Thank you both.

Define a metric on ##\mathbb{R}[x]## for distinct polynomials ##f(x),g(x)## as ##d(f(x),g(x)) = \frac{1}{2^{n}}##, where ##n## is the largest positive integer such that ##x^{n}## divides ##f(x)-g(x)##. Equivalently, ##n## is the multiplicity of the root ##x=0## of ##f(x)-g(x)##. Set...

Using (a) as an example, if m>n, then f_{m} and f_{n} would be the same (both \frac{1}{\sqrt{x}} ) on \frac{1}{n+1} \leq x \leq 1 and different (f_{m}=\frac{1}{\sqrt{x}} but f_{n}=0) on \frac{1}{m+1}\leq x < \frac{1}{n+1}. So \displaystyle f_{m}-f_{n} = \begin{cases} \frac{1}{\sqrt{x}} &...

Homework Statement Determine whether or not the following sequences of real valued functions are Cauchy in L^{1}[0,1]: (a) f_{n}(x) = \begin{cases} \frac{1}{\sqrt{x}} & , \frac{1}{n+1}\leq x \leq 1 \\ 0 & , \text{ otherwise } \end{cases} (b) f_{n}(x) = \begin{cases} \frac{1}{x} & ...

Thank you again, I will keep this in mind.

Ah yes, I think it is clear now. Thank you. Now I can proceed on what I'm supposed to do (proving p and q are independent of choice of basis).

So I break this up\displaystyle Q(\mathbf{v}) = \sum_{i=1}^{n}z_{i}^{2}B(\mathbf{v}_{i},\mathbf{v}_{i}) = z_{1}^{2}B(\mathbf{v}_{1},\mathbf{v}_{1}) + z_{2}^{2}B(\mathbf{v}_{2},\mathbf{v}_{2}) + z_{3}^{2}B(\mathbf{v}_{3},\mathbf{v}_{3}) + \dots + z_{n-1}^{2}B(\mathbf{v}_{n-1},\mathbf{v}_{n-1}) +...

The question comes out of a corollary of this theorem: Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...

Ah, so the problem then becomes to show for all \epsilon>0 there exists N\in\mathbb{N} such that \left| \frac{ (n+1)(n+2) }{2^{n-1}} - \frac{ (m+1)(m+2) }{2^{m-1}} \right| < \epsilon for all n,m \geq N. Once this is proved, (s_{n}) is Cauchy, so it must converge, thus the series converges.

If I'm reading it right, that difference is a set up for Cauchy criteria.

My original attack on the problem was to analyze the product first to see it it converged. I haven't been able to make a conclusion about the series yet. I know that the problem I should be looking at is, \sum_{i=1}^{\infty}\left( \prod_{k=1}^{i} \left( \frac{1}{2} + \frac{1}{k} \right) \right)...

Expanding the partial product, I found that p_{n} = \left(\frac{1}{2} + 1\right)\cdot\left(\frac{1}{2} + \frac{1}{2}\right) \dots \cdot \frac{(n+1)(n+2)}{2^{n-1}}. Fortunately this goes to 0 in the limit.

The original problem was to prove that \sum a_{n} converges, given that a_{n+1} = \left( \frac{1}{2} + \frac{1}{n} \right)a_{n} , with a_{1}=1. After messing around with this, I came up with the formula in my original post. Sorry, I was referring to jargon used here...

Homework Statement Show that the sequence of partial sums s_{n} = 1+\sum_{i=1}^{n} \left(\prod_{k=1}^{i}\left( \frac{1}{2} + \frac{1}{k}\right)\right) converges, with n\in \mathbb{N}\cup \{0\} Homework EquationsThe Attempt at a Solution [/B] So we want to find \lim_{n\to\infty} s_{n} =...

Homework Statement Let P(W) be a projective space whose dimension is greater than or equal to 2 and let three non-colinear projective points, [v_{1}],[v_{2}],[v_{3}]\in P(W) . Prove that there is a projective plane in P(W) containing all three points. Homework EquationsThe Attempt at a...