Mary Boas attempts to explain this by pointing out that the situation cannot arise because charges will have to be placed individually, and in an order, and that order would represent the order we sum in. That at any point the unplaced infinite charges would form an infinite divergent series...
I had thought it would be failure of structural stability since in structural stability qualitative behavior of the trajectories is unaffected by small perturbations, and here, even tiny deviations using ##h## values resulted in huge effects. However, apparently that's not the case, and I'm not...
Let:
##\nabla## denote dell operator with respect to field coordinate (origin)
##\nabla'## denote dell operator with respect to source coordinates
The electric field at origin due to an electric dipole distribution in volume ##V## having boundary ##S## is:
\begin{align}
\int_V...
I have been struggling with a problem for a long time. I need to solve the second order partial differential equation
$$\frac{1}{G_{zx}}\frac{\partial ^2\phi (x,y)}{\partial^2 y}+\frac{1}{G_{zy}}\frac{\partial ^2\phi (x,y)}{\partial^2 x}=-2 \theta$$
where ##G_{zy}##, ##G_{zx}##, ##\theta##...
Homework Statement
- Given a bounded sequence ##(y_n)_n## in ##\mathbb{C}##. Show that for every sequence ##(x_n)_n## in ##\mathbb{C}## for which the series ##\sum_n x_n## converges absolutely, that also the series ##\sum_n \left(x_ny_n\right)## converges absolutely.
- Suppose ##(y_n)_n## is...
I tried to derive an equation for one sample mean to converge to another sample mean within a 95% confidence interval, but I know I am wrong. Can someone tell me what I did wrong, and what is the correct formula?
Suppose:
##\hat{x_1},\hat{\sigma_1},N## are a sample mean, standard deviation...
For a series to be convergent,it must have a finite sum,i.e.,limiting value of sum.As the sum of n terms approaches a limit,it means that the nth term is getting smaller and tending to 0,but why is not the converse true?Should not the sum approach a finite value if the nth term of the series is...
Homework Statement
(FYI It's from an Real Analysis class.)
Show that $$\int_{0}^{\infty} (sin^2(t) / t^2) dt $$ is convergent.
Homework Equations
I know that for an integral to be convergent, it means that :
$$\lim_{x\to\infty} \int_{0}^{x} (sin^2(t) / t^2) dt$$ is finite.
I can also use...
If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?
Hi Physics Forums,
I have a problem that I am unable to resolve.
The sequence ##\{\mathrm{sinc}^n(x)\}_{n\in\mathbb{N}}## of positive integer powers of ##\mathrm{sinc}(x)## converges pointwise to the indicator function ##\mathbf{1}_{\{0\}}(x)##. This is trivial to prove, but I am struggling to...
Homework Statement
Determine whether the following series converge, converge conditionally, or converge absolutely.
Homework Equations
a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity)
b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity)
c) ∑k×sin(1+k^3)/(k + ln(k))...
Homework Statement
I have to prove that the improper integral ∫ ln(x)/(1-x) dx on the interval [0,1] is convergent.
Homework Equations
I split the integral in two intervals: from 0 to 1/2 and from 1/2 to 1.
The Attempt at a Solution
The function can be approximated to ln(x) when it approaches...
Homework Statement
Test the series for convergence or divergence
##1/2^2-1/3^2+1/2^3-1/3^3+1/2^4-1/3^4+...##
Homework Equations
rn=abs(an+1/an)
The Attempt at a Solution
With some effort I was able to figure out the 'n' th tern of the series
an =
\begin{cases}
2^{-(0.5n+1.5)} & \text{if } n...
Homework Statement
I'm trying to solve Laplace's equation numerically in 3d for a charged sphere in a big box. I'm using Comsol, which solves using the finite elements method. I used neumann BC on the surface of the sphere, and flux=0 BC on the box in which I have the sphere. The result does...
Homework Statement
Show that ##\sum_{k=2}^\infty d_k## converges to ##\lim_{n\to\infty} s_{nn}##.
Homework Equations
I've included some relevant information below:
The Attempt at a Solution
So far I've managed to show that ##\sum_{k=2}^\infty |d_k|## converges, but I don't know how to move...
In the textbook I have (its a textbook for calculus from my undergrad studies, written by Greek authors) some times it uses the lemma that
"for any irrational number there exists a sequence of rational numbers that converges to it",
and it doesn't have a proof for it, just saying that it is a...
Homework Statement
Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is:
The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞
The question is also attached as a .png file
2. Homework Equations
The...
I'm trying to expand the following using Newton's Generalized Binomial Theorem.
$$[f_1(x)+f_2(x)]^\delta = (f_1(x))^\delta + \delta (f_1(x))^{\delta-1}f_2(x) + \frac{\delta(\delta-1)}{2!}(f_1(x))^{\delta-2}(f_2(x))^2 + ...$$
where $$0<\delta<<1$$
But the condition for this formula is that...
I am using the static structural module of ANSYS workbench to do a simulation. In my model, there is a gear and a spring which presses against the gear, moves along it and pushes it to turn counterclockwise. These two objects are in frictional contact. In my calculation, I always have the...
Homework Statement
Homework Equations
[/B]
Definition: A sequence X_1,X_2,\dots of real-valued random variables is said to converge in distribution to a random variable X if \lim_{n\rightarrow \infty}F_{n}(x)=F(x) for all x\in\mathbb{R} at which F is continuous. Here F_n, F are the...
Homework Statement
I know that ∑n=1 to infinity (sin(p/n)) diverges due using comparison test with pi/n, despite it approaching 0 as n approaches infinity.
However, an alternating series with (-1)^n*sin(pi/n) converges. Which does not make sense because it consists of two diverging functions...
Dear experts,
I´m performing a non-linear buckling analysis under ANSYS Mechanical APDL (v14.5) using an input file that processes the last converged step to generate some etable output.
When run in GUI everything goes fine: the non-linear buckling analysis is performed until it becomes...
Homework Statement
I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent.
Σ(n3/3n
Σk(2/3)k
Σ√n/1+n2
Σ(-1)n+1*n/n^2+9
Homework Equations
Comparison Test
Ratio Test
Alternating Series Test
Divergence Test, etc
The Attempt at a...
Homework Statement
Hi
I am looking at the proof attached for the theorem attached that:
If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2##
where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##.
For any integer ##r \geq 0 ## :
##\Omega_r := {mw_1+nw_2|m,n \in...
Homework Statement
Consider the space ##([0, 1], d_1)## where ##d_1(x, y) = |x-y|##. Show that there exists a sequence ##(x_n)## in ##X## such that for every ##x \epsilon [0, 1]## there exists a subsequence ##(x_{n_k})## such that ##\lim{k\to\infty}\space x_{n_k} = x##.
Homework Equations
N/A...
1. The problem:
Ive been all afternoon struggling with this doubt. Its a bit more teoric than the rest of the exercices i did and i just cant seem to get around it so here it goes ...
Consider a sample consisting of {y1,y2,...,yk} realisations of a random variable Y, and let S(k) denote the variance of the sample as a function of its size; that is
S(k)=1/k( ∑ki=1(yi−y¯)2)
for y¯=1/k( ∑ki=1 yi)
I do not know the distribution of Y, but I do know that S(k) tends to zero as k...
I just took a calc 2 test and got 3/8 points on several problems that asked you to show convergence or divergence. The reason being that I didn't use the correct test of convergence? The answer was right, if you get to the point where you know the series converges, then why does it matter which...
Homework Statement
Determine which of the sequences converge or diverge. Find the limit of the convergent sequences.
1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)]
Homework Equations
[/B]
a1=first term, a2=second term...an= nth term
The Attempt at a Solution
a) So I found the first couple of...