Cauchy Definition and 381 Threads

Homework Statement Prove that if {a_{n}} is a sequence of rational numbers such that {a_{n+1}} > {a_{n}} for all n \in \textbf{N} and there exists an M\in \textbf{Q} such that {a_{n}} \leq M for all n \in \textbf{N}, then {a_{n}} is a Cauchy sequence of rational numbers.Homework Equations Do...

So... I want to find the Cauchy sum of the Taylor polynomial of \exp x \sin x. I know how to do this with maple, which only requires the command taylor(sin(x)*exp(x), x = 0, n). I can also try the good old f(a)+\frac{f'(a)}{1!}(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+\cdots...

Homework Statement Thm: If a sequence is Cauchy than that sequence is bounded. However Take the partial sums of the series (sigma,n->infinity)(1/n). The partial sums form a series which is Cauchy. But the series diverges so the sequence of partial sums is unbounded. Sequence of partial sums...

Homework Statement In my book (Classical Analysis by Marsdsen & Hoffman), they use the monotone bounded sequence property as the completeness axiom. That is to say, they call complete an ordered field in which every bounded monotone sequence converges and they argue that there is a unique (up...

Would someone care explaining to me what a Cauchy surface is? thanks

I have generated 2 columns of normal random variables, Z1 and Z2. Theorectically, Z1/Z2 will follow a Cauchy distribution. The question is, how do I construct a probability plot to show that indeed it is a Cauchy distribution? I tried the follow procedure: -Sort the Z1/Z2 -Rank them and...

Homework Statement If p does not divide a, show that a_n=a^{p^{n}} is Cauchy in \mathbb{Q}_p. The Attempt at a Solution We can factor a^{p^{n+k}}-a^{p^n}=a^{p^n}(a^{p^{n+k}-1}-1). p doesn't divide a^{p^n} so somehow I must show that a^{p^{n+k}-1}-1 is divisible by larger and larger powers of...

Homework Statement For r=1,3,5 compute the following integral: Integral over alpha (e^(x^2)/(x^2-6x)dx Alpha(t) = 2+re^(it) from 0 to 2pi Homework Equations Cauchy Integral Formula: f(z) = 1/(2ipi)Integral over Alpha(f(x)/(x-z)dx) The Attempt at a Solution For r = 1...

Homework Statement q(n) = Sum(from k=1 to n) 1/n! Exercise 3: Prove that {q(n)}n(forall)Ns is a cauchy sequence. Homework Equations none. The Attempt at a Solution So many attempts at a solution. I know that a sequence is a cauchy sequence if for all epsilons greater than...

The cauchy Riemann relations can be written: \frac{\partial f}{\partial \bar{z}}=0 Is there an 'easy to see reason' why a function should not depend on the independent variable [itex]\bar{z}[/tex] to be differentiable?

I tried to expand the [SUM{[X sub k + Y sub k]^2}]^1/2 term but I am stuck there.

Lets say we have: (a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n})^{2} \leq (a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2})(b_{1}^{2} + b_{2}^{2} + ... + b_{n}^{2}) . Let A = a_{1}^{2} + a_{2}^{2} + ... + a_{n}^{2} , B = a_{1}b_{1} + a_{2}b_{2} + ... + a_{n}b_{n}, C = b_{1}^{2} + b_{2}^{2} + ... +...

I was doing a Fourier Transform Integral, and was wondering if it would be legitimate for me to choose a semicircle C[SIZE="2"]R on the lower half-plane below the real axis rather than choosing a semicircle C[SIZE="2"]R on the upper half-plane above the real axis. I would expect it to be valid...

Hi again. Can somebody help me out with this question? "\int_{C_1(0)} \frac {e^{z^n + z^{n-1}+...+ z + 1}} {e^{z^2}} \,dz Where C_r(p) is a circle with centre p and radius r, traced anticlockwise." I'd be guessing that you have to compare this integral with the Cauchy integral formula...

Hi, Here's the question: Show that if {x_n} is a cauchy sequence of points in the metric space M, and if {x_n} has a subsequence which converges to x \in M, Prove that x_n itself is convergent to x. Now, I have proved this as follows..I didn't put in all of the details... Let {x_n_k} be the...

I have a function 2-z^2-2\cos z, which has a zero at z=0. I have determined the Maclaurin series for f: \sum_{j=2}^\infty(-1)^{j-1}\frac{2z^{2j}}{(2j)!}, and now I have to determine the coefficients a_{-j},~\forall j>0, in the Laurent series for a function h, which is defined as...

How would one prove that the sum of 2 cauchy sequences is cauchy? I said let e>0 and take 2 arbitrary cauchy sequences then |Sn - St|<e/2 whenever n,t>N1 and |St - Sm|<e/2 whenever t,m >N2. So |Sn - Sm|=|Sn - St + St - Sm|<= |Sn - St|+|St - Sm|< e/2 + e/2 <= e So n,m>max{N1, N2}...

Question: Prove that if a Cauchy sequence x_1, x_2,... of rationals is modified by changing a finite number of terms, the result is an equivalent Cauchy sequence. All the math classes I have taken previously were computational, and my textbook contains almost no definitions. So, I...

Here's the problem statement: Prove that x_1,x_2,x_3,... is a Cauchy sequence if it has the property that |x_k-x_{k-1}|<10^{-k} for all k=2,3,4,.... If x_1=2, what are the bounds on the limit of the sequence? Someone suggested that I use the triangle inequality as follows: let n=m+l...

Question Consider the sequence \{p^n\}_{n\in\mathbb{N}}. Prove that this sequence is Cauchy with respect to the p-adic metric on \mathbb{Q}. What is the limit of the sequence?

hello all I found this rather interesting suppose that a sequence {x_{n}} satisfies |x_{n+1}-x_{n}|<\frac{1}{n+1} \forall n\epsilon N how couldn't the sequence {x_{n}} not be cauchy? I tried to think of some examples to disprove it but i didnt achieve anything doing that, please...

I need help on trying to prove that every subsequence of a cauchy sequence is a cauchy sequence

Im in need of some guidance. No answers, just guidance. :smile: Question. Let (x_m) be a Cauchy sequence in an inner product space, show that \left\{\|x_n\|:n=1,\dots,\infty\right\} is bounded. proof From the definition we know that all convergent sequences are Cauchy...

The question calls for using Cauchy's integral formula to compute the integral for Int.c z/[(z-1)(z-3i)] dz, assuming C is the loop |z-1|=3. Taking z = 1 and f(z) = z/(z-3i), I came up with (2pi*i)/(1-3i), which seems like it could be simplified, but I'm not sure how.

I am intriqued by a recent series of three papers on black holes: http://www.arxiv.org/abs/gr-qc/0411060 Title: The river model of black holes Authors: Andrew J. S. Hamilton, Jason P. Lisle (JILA, U. Colorado) http://www.arxiv.org/abs/gr-qc/0411061 Title: Inside charged black holes I...

I know that if the series of (a)n (n is a subscript) converges, then the lim (a)n=0. How can I show that if the series of (a)n converges, then lim n(a)n=0? Or rather if a1 +a2 +a3 +...+an=0, then lim n*(a)n=0? Not sure how to show this, but I know the proof involves the cauchy criterion...

Let {an}(n goes from 1 to infinity) be a sequence. For each n define: sn=Summation(j=1 to n) of aj tn=Summation(j=1 to n) of the absolute value of aj. Prove that if {tn}(n goes from 1 to infinity) is a Cauchy sequence, then so is {sn}(n goes from 1 to infinity). I started this...

Ok, I am told in a complex analysis book that the gradient squared of u is equal to the gradient squared of v which is equal to 0. We know the derivate of w exists, and w(z)=u(x,y) + iv(x,y) Thus the Cauchy Riemann conditions must hold. (When I use d assume that it refers to a partial...

Is it possible to solve for an E field from a charge density function using the Cauchy Integral Formulas from complex variables? Cauchy Integral Formula about a closed loop in the complex plane (Integral[f[z]/ (z-z0)^(n+1)dz = 2 pi i /n! d^n f(z0)/dz ]) that is the n derivative of f with...

Hi, I really need some help in sovling this proof! Prove the Cauchy Mean Value Theorem: If f,g : [a,b]->R satisfy f continuous, g integrable and g(x)>=0 for all x then there exists element c is a member of set [a,b] so that int(x=b,a)f(x)g(x)dx=f(c)int(x=b,a)g(x)dx. Thanks for your help :D

Did anyone else hear about this new development? Apparently if a black hole has a steady influx of matter/energy, it may not develop a singularity which brings about infinite tidal distortion, but it could bring about a 'gentler' cauchy horizon singularity that could be possible to traverse...