Convergence Definition and 1000 Threads

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 [FONT="Arial Black"]OrCAD PSpice ebooks, application notes and tutorials and how to resolve its convergence issues in [FONT="Arial Black"]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...

Okay, this question straddles math, physics, biology, and chemistry, so I'm not sure if this is the correct forum to post it in, but I was using mostly chemistry knowledge to solve it, so I would guess that may be the correct method. Anyway, it's not homework, but it is similar to many homework...

I'm trying to refresh my knowledge of Calc II, and I'm going through improper integrals right now. The problem I am trying to solve is: For which numbers p\geq0 does \int_0^\infty \frac{e^{-x}}{x^{p}} converge? Justify your answer. So far, I've split up the integral into two halves (0 to 1...

Homework Statement This is not so much an entire problem I need help with but just a part. It is a power series where after you do the ratio test, you end up with |4x^(2)| < 1, so |x^(2)| < 1/4. Since the radius of convergence is |x-a| < R, I end up with -1/4 < x^(2) < 1/4, but because...

I have a question that I've been pondering recently. As far as I can tell, it's original to the boards or at least hasn't been discussed in a long time so I think it's fair to start a new topic. This concerns initial bases for thought. It would seem that both language and mathematics are the...

Homework Statement Find the Summation Notation and Radius of Convergence of this series. 5, x, 10, x, ... The Attempt at a Solution I don't know how did they come up with that equation.. But the summation seems right.. Can anyone tell me how did they arrive with that equation? I've tried...

Homework Statement Test for convergence or divergence. Give a reason for your decision. Homework Equations \sum_{i=1}^{\infty} \frac{\sqrt{2n-1} \log{(4n + 1)}}{n(n + 1)} The Attempt at a Solution I've tried to compare it to the series \sum_{i=1}^{\infty} \frac{\sqrt{2n-1} \log{(4n +...

Hi, I was always troubled by the relationships between these modes of convergence (L^1, L^2, and L^{\infty} convergences, to be precise), so I took some books and decided to establish some relations between them. For some, I succeeded, for others I did not. Here's what I did so far: If I...

Homework Statement given an = ( -1 )(n+1) / \sqrt{n} determine if the infinite series is Absolutely Convergent, Conditionally Convergent, or Divergent. Homework Equations I hope I have these theorems down correctly, please correct me if I'm wrong. If \Sigma|an| is Convergent then...

This problem has been bothering me for some time. Any thoughts or insights are greatly appreciated. Consider a function, f, with continuous third derivative on [-1,1]. Prove that the series \sum^{\infty}_{n=1} (nf(\frac{1}{n})-nf(-\frac{1}{n}) - 2\frac{df}{dn}(0)) converges. Thanks in...

I've been having some trouble understanding how to determine if a sequence is divergent or convergent. For example an = cos(2/n) I know if I take the limit as n ->\infty then I will get 1. So the sequence has a limit but does having a limit mean that the sequence is convergent.

Definition: Let X1,X2,... be a sequence of random variables defined on a sample space S. We say that Xn converges to a random variable X in probability if for each ε>0, P(|Xn-X|≥ε)->0 as n->∞. ==================================== Now I don't really understand the meaning of |Xn-X| used in...

Homework Statement Let z,p,q \in \mathbb{C} be complex parameters. Determine that the Gamma and Beta integrals: \displaystyle \Gamma (z) = \int_0^{\infty} t^{z-1} e^{-t}\;dt \displaystyle B(p,q) = \int^1_0 t^{p-1} (1-t)^{q-1}\;dt converge absolutely for \text{Re}(z)>0 and p,q>0...

I have a question regarding the following definition of convergence on manifold: Let M be a manifold with atlas A. A sequence of points \{x_i \in M\} converges to x\in M if there exists a chart (U_i,\phi_i) with an integer N such that x\in U_i and for all k>N,x_i\in U_i \phi_i(x_k)_{k>N}...

Homework Statement If the absolute value of a sequence, an converges to absolute value of A, does sequence, an necessarily converge to A? Homework Equations convergence: a sequence { an}n=1-->infinity, converges to A є R (A is called the limit of the sequence) iff for all є > 0, there...

Homework Statement Sum 1/n(n+1) * (sin(x))^n . Show this converges for all x in the reals. Find with proof an interval on which it determines a differentiable function of x together with an expression of its derivative in terms of standard functions. Homework Equations The...

Homework Statement Find what range of values of x the infinite sum of sin2n(x) and infinite sum 2nsin2n-1(x) converge and find an expression for their sums, carefully justifying your answers. The Attempt at a Solution I used cauchys root testand basically got that the first sum...

Hi, Here is my question: Given that X_n\xrightarrow{\mathcal{D}}Z as n\rightarrow\infty where Z\sim N(0,1). Can we conclude directly that \lim_{n\rightarrow\infty}P(|X_n|\leq u)=P(|Z|\leq u) where u\in (0,1)? Is this completely trivial or requires some proof? Also what is the differences...

Homework Statement Determine whether 1/n! diverges or converges. So far, we have only learned the comparison tests, p-series, geometric series, divergence test, and integral test, so I can only use these tests to prove it. Homework Equations N/a The Attempt at a Solution I...

Homework Statement Consider the infinite series (1/n) * (xn) where x is a real noumber. Find all numbers x such that i) the series converges, ii) series converges absolutely iii) diverges to + infinity iiii) does not converge. 2. The attempt at a solution For this, i know it...

Homework Statement I am not really good with Series so I having a hard time with these problems. http://img835.imageshack.us/img835/858/img1257d.jpg Homework Equations The Attempt at a Solution The part I am stuck is where I highlighted. The first question: The whole thing is squared so I...

Homework Statement The implicit Euler method is yn = yn-1 + hf(xn,yn). Find the local truncation error and hence show that the method is convergent. Homework Equations The Attempt at a Solution I found the error to be ln = (-h2/2)y''(xn-1) + O(h3). For convergence I am up to...

Homework Statement I need to examine convergence of series with a term Un given below. The solution is given, but I can't understand what happens between row 2 and row 3. What kind of operation is that, does it have something in common with Taylor series expansion...

Absolute Convergence, Conditional Convergence or divergence... Homework Statement \sum_{n=1}^{\infty} \frac {(-2)^{n}}{n^{n}} Homework Equations \lim_{n \rightarrow \infty} | \frac {a_{n+1}}{a_n}| < 1 \;\; absolute\; convergence \lim_{n \rightarrow \infty} | \frac...

Homework Statement If the series \sum_{n=1}^{\infty}x_n converges absolutely, and the sequence (y_n)_n is bounded, then the series \sum_{n=1}^{\infty}x_ny_n converges.Homework Equations Definitions and theorems relating to series and convergence.The Attempt at a Solution If the sequence y_n is...

Homework Statement Find the interval of convergence: \sum _{n=1}^{\infty } \frac{(-1)^n (x+2)^n}{3^n\sqrt{n}} Homework Equations The Attempt at a Solution \lim_{n\to \infty } |\frac{(x+2)^{n+1}}{3^{n+1}\sqrt{n+1}}*\frac{3^n\sqrt{n}}{(x+2)^n}| = \lim_{n\to \infty }...

So I've seen the distinction one makes in case of infinite-dimensional Hilbert spaces. Weak convergence versus strong convergence of sequences. I cannot think of an example of sequence of vectors in L^2(R) which converges with respect to the scalar product, but not with respect to the norm...

Homework Statement I need help finding the interval of convergence for f(x) = 3/(1-x^4). I think that the summation would be \Sigma 3 (x^4n) from n=0 to infinity, but I'm not sure how to get the interval of convergence. Homework Equations f(x) = 3/(1-x^4) The Attempt at a Solution...

Homework Statement Prove that the series \sum_{n=0}^\infty e^{-n^2x^2} converges uniformly on the set \mathbb{R}\backslash\ \big] -\epsilon,\epsilon\big[ where \epsilon>0Homework Equations n/aThe Attempt at a Solution I have tried using Weierstrass M-test but I have not been able to find a...

I was thinking of how ( 1 + (1/n) ) ^ n converges to e and I am aware of how if it is raised to some an, then it converges to e^a. If i recall if the form ( 1 + (a/n) ) ^ n converges to ae? I was hoping someone could tell me how to deal with ( 1 + (1/n^2) ) ^ n? Thanks!

Homework Statement Sum ln n/(ln(ln n)) n=3..infinity? Im pretty sure it diverges and I am pretty sure to use the limit test but i just don't know what to compare this sum to. Would 1/n be ok. Do i have to justify why they are similar? ANy help would be nice thanks. Homework Equations...

Homework Statement For each of the following sequences (fn), find the function f such that fn --> f. Also state whether the convergence is uniform or not and give a reason for your answer. Homework Equations a.) fn(x) = 1/xn for x greater than or equal to 1 b.) f[SUB]n[SUB](x) =...

Homework Statement use dirichlet test to determine if the series converges 1-1/2-1/3+1/4+1/5-1/6-1/7+... Homework Equations The Attempt at a Solution I have broken up the series into two different series the first series I have is 1+1/4+1/8+1/12+... and the second series I...

Homework Statement This isn't really so much a homework problem as me asking a question. The Taylor Series for ln(x) centered at 1 is: sum_[0, infinity] ((-1)^((n+1)*(x-1)^n))/n, then why does it converge for the endpoint x=2, but not x=0 of the interval of convergence? Homework Equations The...