CONvergence is an annual multi-genre fan convention. This all-volunteer, fan-run convention is primarily for enthusiasts of Science Fiction and Fantasy in all media. Their motto is "where science fiction and reality meet". It is one of the most-attended conventions of its kind in North America, with approximately 6,000 paid members. The 2019 convention was held across four days at the Hyatt Regency Minneapolis in Minneapolis, Minnesota.
Homework Statement
X, Y, (X_n)_{n>0} \text{ and } (Y_n)_{n>0} are random variables.
Show that if
X_n \xrightarrow{\text{P}} X and Y_n \xrightarrow{\text{P}} Y then X_n + Y_n \xrightarrow{\text{P}} X + Y
Homework Equations
If X_n \xrightarrow{\text{P}} X then...
Im struggling with the concept of this basic sequence question.
Let x(n) be a sequence such that lim(n->00) (nx(n)) = 0
i.e. it converges to zero...
How could i show that there is an N s.t. for all n≥N : -1 < nx(n) < 1
Any tips would be great.. I don't want an answer.. I want to...
I am given f_n(x)=\frac{nx}{nx+1} defined on [0,\infty) and I have that the function converges pointwise to 0 \ \mbox{if x=0 and} 1\ \mbox{otherwise}
Is the function uniform convergent on [0,1] ?
No. If we take x=1/n then Limit_{n\rightarrow\infty}|\frac{1/n*n}{1+1/n*n}-1|=0.5...
Homework Statement
\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sqrt{n}+1}{n+1}
Homework Equations
absolute convergence test
The Attempt at a Solution
by book says that the series converges because \sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1} converges
but they don't show how the absolute...
Homework Statement
Find the minimum number required (value of n) for the average deviation of the Fourier Series to fall below 2%
Homework Equations
Use the Uniform Convergence of Fourier Series.
Where Sm is the partial sum of the Fourier Series.
C is constant. Here C is ∏^2
So...
Homework Statement
Given each of the functions f below, describe the set of points at which the Fourier
series converges to f.
b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x)
Homework Equations
Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...
Homework Statement
I need to understand and prove the following: That if a>1 the function diverges, except for a special case x_0= b/(1-a). Then if a=-1 diverges for some cases and converges if x_0 is b/2. Again, not to clear on this.
Homework Equations
lim n →∞...
Homework Statement
This is a homework question for a introductory course in analysis. given that
a) the partial sums of f_n are uniformly bounded,
b) g_1 \geq g_2 \geq ... \geq 0,
c) g_n \rightarrow 0 uniformly,
prove that \sum_{n=1}^{\infty} f_n g_n converges uniformly (the whole...
Homework Statement
I need to calculate the point of divergence for this exponential function :
F(x)= 5.282 * exp ( -0.01726 * x )
may you help me in finding the method to solve such problems ?
Homework Equations
The Attempt at a Solution
Homework Statement
Consider the system
x = \frac{1}{\sqrt{2}} * \sqrt{1+(x+y)^2} - 2/3
y = x = \frac{1}{\sqrt{2}} * \sqrt{1+(x-y)^2} - 2/3
Find a region D in the x,y-plane for which a fixed point iteration
xn+1 = \frac{1}{\sqrt{2}} * \sqrt{1+(x_n + y_n)^2} - 2/3
yn+1 =...
Hello,
I'm pondering over this research question.
Let's suppose you've got a bunch of units which can be colored black or white. They're roaming around 2d grid in random walk. Any time a unit meets with another unit, it has an option to change color. It doesn't have to though, depending...
Hi i have to show that the series 1+2r+r2+2r3+r4+2r5+... converges for r=\frac{2}{3} and diverges for r=\frac{4}{3} using the nth root test.
The sequence \sqrt[n]{a_{n}}comes a bit complicated so i was wondering if I can remove the 1st term a1=1 and show that 2r+r2+2r3+r4+2r5+... converges...
Homework Statement
Use the definition of convergence to prove that lim n→∞ (1/2)^n=0
The definition of convergence says |a_n-L|<ε
Homework Equations
The Attempt at a Solution
As I understand it:
|(1/2)^n-0|<ε
|(1/2)|^n<ε
then I need to solve for n...
The problem statement
Let f:[a,b]→\mathbb{R} be differentiable and assume that f(a)=0 and \left|f'(x)\right|\leq A\left|f(x)\right|, x\in [a,b].
Show that f(x)=0,x\in [a,b].
The attempt at a solution
It was hinted at that the solution was partly as follows. Let a \leq x_0 \leq b. For all x\in...
So a well-known theorem from Lebesgue integration is the dominated convergence theorem. It talks about a sequence f_1,f_2,\ldots of functions converging pointwise to a function f. And if |f_n(x)| \leq g(x) for an integrable function g, then we have \int f_n \to \int f.
But what if we have a...
Homework Statement
The following is a modification of Newton's method:
xn+1 = xn - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn)
Homework Equations
We are supposed to use the following method:
let En = xn + p where p = root → xn = p + En
Moreover, f(xn) = f(p + En) = f(p) +...
Homework Statement
I am asked to comment on the convergence/divergence of three series based on some given information about a power series:
\sum_{n=0}^{\infty}c_nx^n converges at x=-4 and diverges x=6.
I won't ask for help on all of the series, so here's the first one...
Homework Statement
show that x_n converges to x if and only d(x_n, x) converges to 0.
Homework Equations
|x_n - x| < ε for all ε>0
The Attempt at a Solution
well d(x_n,x) converges to 0 if d(x_n,x)<ε
i just don't know how to relate that back to |x_n - x|
Homework Statement
I just got done proving Gauss' test, which is given in the book as:
If there is an N\ge 1, an s>1, and an M>0 such that
\frac{a_{n+1}}{a_n}=1 - \frac{A}{n} + \frac{f(n)}{n^s}
where |f(n)|\le M for all n, then \sum a_n converges if A>1 and diverges if A \le 1.
This...
1. The problem statement:
In what region can we choose x0 and get convergence to the root x = 0 for f(x) = e-1/x^2
Homework Equations
xn+1 = xn - f(xn) / f'(xn)
The Attempt at a Solution
The only thing I've come across is a formula that says |root - initial point| < 1/M where M =...
If you are given part of a period of a Function, what rules would you apply to draw out the full function, so that it converges as quickly as possible as a Fourier series?
thanks
Homework Statement
Determine the behavior of convergence on the unit circle, ie |z| = 1 of:
Ʃ \frac{z^{n}}{n^{2}(1 - z^{n})}
Homework Equations
Obviously this is divergent then z is a root of unity. The question is what happens when z is not a root of unity.
The Attempt at a...
Homework Statement
Find the power series representation for the function f(x)=x/(x^2-3x+2) and determine the interval of convergence.
Homework Equations
The Attempt at a Solution
First I separate into partial fractions 2/(x-2) - 1/(x-1)
2/(x-2) = sum n=0 to infinity (x/2)^n...
I was just watching a television program about gravity today and it got me wondering what gravity was exactly.
Most analogies used to describe gravity are of a heavy ball on a bed sheet. The ball creates a depression in the sheet and objects placed on the sheet will fall in towards the ball...
Homework Statement
I'm asked to specifically use the Ratio Test (formula below) to determine whether this series converges or diverges (if it converges, the value to which it converges is not needed.)
\sum_{n=1}^{\infty}\frac{n}{(e^n)^2}
Homework Equations
Ratio Test:
If a_n is a sequence...
Homework Statement
prove that the series summation from n=3 to infinity of (1/(n*log(n)*(log(log(n))^p)) diverges if 0<p<=1 and converges for p>1.
Homework Equations
The Attempt at a Solution
2^n*a(2^n)= 1/(log(2^n)*(log(log(2^n))^p)). this is similar to the summation from n=2 to...
I know this is like very basic, but my brain just somehow couldn't accept it!
Homework Statement
I don't understand why does the sequence (rn) converges to 0 as n -> infinity when -1<|r|<1
The Attempt at a Solution
i did quite a few ways to convince myself.
Firstly, we know that...
Homework Statement
assume summation of series An converges with all An>0. Prove summation of sqrt(An)/n converges
Homework Equations
The Attempt at a Solution
I Tried using the ratio test which says if lim as n goes to infinity of |Bn+1/Bn|<1 then summation of Bn converges. I let Bn...
Okay, so I didn't really understand the professor when he talked about the speed of convergence of Fourier series. The question is what kind of functions converge faster than what kind of other functions using Fourier series representation. My guess from what I have absorbed is that functions...
Hi,
Here's another question from my analsysi HW. I get that the two sequences are equal but I'm not sure how to write it out. Any help would be great.
Thanks.
Homework Statement
Prove that a sequence {a_n} converges to 0 iff the sequence {\lvert a_n\rvert} converges to 0.
Homework...
Hi,
I'm doing some homework from my analysis class. I honestly have no idea where to start. Any help would be appreciated.
Homework Statement
Let {a_n} be a sequence that converges to 0, and let {b_n} be a sequence. Prove that the sequence a_n b_n converges to 0.
Homework Equations...
1. Prove that n^(1/n) converges to 1.
3. I've attempted to define {a} = n^1/n - 1 and have shown, using the binomial formula, that n=(1+a)^2>=1+[n(n-1)/2]*a^2. I think I'm on the right track but don't know how to bring this back to the original problem to prove convergence even after staring...
Homework Statement
I am asked to determine whether a series converges, and if so, to provide its sum.
The problem is:
\sum_{n=1}^{\infty}(-3)^{n-1}4^{-n}
Homework Equations
- I know that if the limit of the sequence as n->inf is finite, then the series converges at that limit.
- I also...
Homework Statement
Prove that lim_{n} p_{n}= p iff the sequence of real numbers {d{p,p_{n}}} satisfies lim_{n}d(p,p_{n})=0
Homework Equations
The Attempt at a Solution
I think I can get the first implication. If lim_{n} p_{n}= p, then we know that d(p,p_{n}) = d(p_{n},p) <...
Homework Statement
I have to find the order of convergence of the following sequence
b_n = \left( \frac{5}{6} \right)^{n^2}
I have numerically tested that it has to be a real number between 1 and 2, but I can't find it exactly.
I also have this doubt: does every sequence have an...
I decided to put my attempt at a solution before the question, because the "solution" is what my question is about.
Homework Statement
Find the rate of convergence for the following as n->infinity:
lim [sin(1/n^2)]
n->inf
Let f(n) = sin(1/n^2) for simplicity.
2. The attempt at a...
Homework Statement
let (An) be a sequence in R with |summation from n=1 to infinity(An)|< infinity. Prove lim as n goes to infinity of ((A1 +2A2+...+nAn)/n) = 0
Homework Equations
The Attempt at a Solution
I think |summation from n=1 to infinity(An)|< infinity means the summation...
Homework Statement
Let b_n be a bounded sequence of nonnegative numbers. Let r be a number such that 0 \leq r < 1.
Define s_n = b_1*r + b_2*r^2 + ... + b_n*r^n, for all natural numbers n.
Prove that {s_n} converges.
Homework Equations
Sum of first n terms of geometric series = sum_n...
Homework Statement
A function f(x) is given as follows
f(x) = 0, , -pi <= x <= pi/2
f(x) = x -pi/2 , pi/2 < x <= pi
determine if it's Fourier series (given below)
F(x)=\pi/16 + (1/\pi)\sum=[ (1/n^{2})(cos(n\pi) - cos(n\pi/2))cos(nx)
-...
Homework Statement
f(x) = 5, -pi <= x <= 0
f(x) = 3, 0 < x <= pi
f(x) is the function of interest
Find the x-points where F(x) fails to converge
to f(x)
Homework Equations
F(x) = f(x) if f is continuous at x\in(-L,L)
F(x) = 0.5[ f(x-) + f(x+) ] if f is...
Homework Statement
Let \left (X,d \right) be a metric space, and let \left\{ x_n \right\} and \left\{ y_n \right\} be sequences that converge to x and y. Let \left\{ z_n \right\} be a secuence defined as z_n = d(x_n, y_n). Show that \left\{ z_n \right\} is convergent with the limit d(x,y)...
Homework Statement
For f(z) = 1/(1+z^2)
a) find the taylor series centred at the origin and the radius of convergence.
b)find the laurent series for the annulus centred at the origin with inner radius given by the r.o.c. from part a), and an arbitrarily large outer radius.
Homework...
Homework Statement
let (Xn) be a sequence in R given by X1=1 and Xn+1=1/(3+Xn) for n>=2. prove Xn converges and find the limit.
Homework Equations
The Attempt at a Solution
well i think using the monotone convergence theorem would help but i would have to prove that the sequence...
Hi
any idea what is the most relevant website to find out OrCAD PSpice ebooks, application notes and tutorials and how to resolve its convergence issues in Switch Mode Power Supplies simultions.
Please be precise and quick. :cool:
I know that the following serie converges to 2 (did in excel), still I would like to know how i can prove it step by step it.
∞
∑ n/2^n
n=1
I tried (n+1)/(2^(n+1))/(n/2^n) still I'm finding 1/2, not the 2.
Any thoughts?
Is it true that a cauchy sequence of continuous functions defined on the whole real line converges uniformly to a continuous function?
I thought this was only true for functions defined on a compact subset of the real line.
Am I wrong?
Homework Statement
The Attempt at a Solution
I'm having some trouble getting my head around these 3 problems. Any ideas on how to approach them are welcome.
Homework Statement
Determine if the series
inf
Sigma n/(2n+1)
n=1
converges
Homework Equations
The Attempt at a Solution
When i did this I originally I thought I would just apply the divergence test
lim n/(2n+1) =/= 0
n->inf
there fore I thought by the...
Hi,
I'm studying infinite series and am really struggling with memorizing all the tests for convergence in my book, there's like 10 of them. I don't think I'm going to be successful in memorizing all of them. I will never be asked in my course to use a specific test to determine convergence...