Convergence Definition and 1000 Threads

In Spivak's Calculus, there is a theorem relating the derivative of the limit of the sequence {fn} with the limit of the sequence {fn'}. What I don't like about the theorem is the huge amount of assumptions required: " Suppose that {fn} is a sequence of functions which are differentiable on...

Homework Statement Does \sum _{ n=1 }^{ \infty }{ \frac { { \alpha }^{ n }{ n }! }{ { n }^{ n } } } converge \forall |\alpha |<e and if so, how can I prove it? Homework Equations { e }^{ x }=\sum _{ n=0 }^{ \infty }{ \frac { { x }^{ n } }{ n! } } The Attempt at a Solution...

Homework Statement The Attempt at a Solution Obviously brackets mean something other than parentheses because .5[0 + 0] ≠ .5

Homework Statement This series is what dictates the graph above. The Attempt at a Solution I don't understand what's going on. If they're using the series that i pasted below then why aren't they multiply each value in the brackets by -2/pi? I also don't get why terms...

Homework Statement Hi. I'm trying to solve a serie: Ʃ1∞2n+1*(n+1)! / (n+1)n+1 Homework Equations The Attempt at a Solution I tried solving it with Cauchy's method, but it failed. I also tried using d'alembert criterion, which game me the answer 2, so it should be divergent. However in the...

I have the following problem: prove that the sequence e^{inx} tends to 0, in the sense of distributions, when n\to \infty. Here it is how I approached the problem. I have to prove this: \lim \int e^{inx}\phi(x)\,dx=0 , where \phi is a test-function. I changed variable: nx=x' and got...

Note: This is not strictly a homework problem. I'm just doing these problems for review (college is out for the semester) - but I wasn't sure if putting them on the main part of the forum would be appropriate since they are clearly lower-level problems.(Newbie) Homework Statement The...

I had a bit of trouble in testing series like this for convergence $$\sum_{ n=1 }^{ \infty } \frac { 1 }{ 2n+1 } $$ If by the comparison test, ##\frac{ 1 }{ 2n+1 } < \frac{ 1 }{ 2n }## for all of n>0, and ##\lim_{ n \rightarrow \infty} \frac{ 1 }{ 2n }## =0, then the series should be...

\sum_{r=1}^{\infty} \frac{r!}{3^{r^{2}}} My solution: \frac{3^{r^2}}{r!} > r^2 So \frac{r!}{3^{r^2}} < \frac{1}{r^2} So as 1/r^2 converges, it converges by comparison test. This was in my exam today, I messed up a lot leading up to it. But the question said I could use any test in...

Homework Statement (a)\;\sum^{\infty}_{n=1}\frac{n-5}{n^2}\;(solved) (b)\;\sum^{\infty}_{r=1}\frac{2r}{1+r^2}\;(solved) (c)\;\sum^{\infty}_{n=1}\frac{\cos^4 nx}{n^2}\;(solved) (d)\;\sum^{\infty}_{n=1}\frac{3^r+4^r}{4^r+5^r} (e)\;\sum^{\infty}_{r=1}\frac{r^r}{r!}\;(solved)...

Homework Statement I'm trying to determine if Ʃ 1/(3^ln(n)) converges. Homework Equations The Attempt at a Solution The preliminary test isn't of any help since lim n→∞ an = 0. I tried the integral test but I couldn't integrate the function, and I don't think it's the best...

Homework Statement Find the radius of convergence of the Taylor series at 0 of this function f(z) = \frac{e^{z}}{2cosz-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far: First, I tried to re-write it as a Laurent series to find...

So I have a definition; Xn n=1,2... is a sequence of random variables on ( Ω,F,P) a probability space, and let X be another random variable. We say Xn converges to X almost surely (P-a-s) iff P({limn →∞ Xn=X}C) = 0 It then goes on to say that checking this is the same as checking limm...

Homework Statement Find the Divergence or Convergence of the series \sum^{∞}_{n=1}\frac{2n^2+3n}{\sqrt{5+n^5}} Homework Equations Ratio Test, Comparison Test, Limit Comparison Test, Integral test etc. The Attempt at a Solution This question was on my final exam and the only question of...

Homework Statement Find the radius of convergence of the Taylor series at z = 1 of the function: \frac{1}{e^{z}-1} Homework Equations The Attempt at a Solution Hi everyone, Here's what I've done so far. Multiply top and bottom by minus 1 to get: -1/(1-e^z) And then...

I have a pesky problem, I have this function of time, S(t) and I'm trying to find how far to evaluate S (its an expensive process and must be done for finite t=time). Essentially, I want to measure S until dS/dt ≈ 0. But my current criteria is making the computation itself inefficient not to...

Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. ##\sum _{n=1}\left( -1\right) ^{n}\dfrac {n} {n^{2}+1}## the sum goes to infinity. Homework Equations Theorem for absolute convergence. Test for divergence The Attempt...

Homework Statement ##\sum _{n=1}^{\infty }\dfrac {\left( -3\right) ^{n}} {n^{3}}## According to Wolfram Alpha the series diverges by the Limit Comparison Test, but I remember that the limit comparison only works with series greater than zero. How is this possible? Homework Equations...

Homework Statement Ʃ[(-1)^n (cosn)^2]/√n The Attempt at a Solution i don't have the slightest clue where to start

Determine whether the series converges or diverges: \sum ln k/ k3 now I said that this series converges by the comparison test, using ln k / k since I know that goes to 0 Would that be the right logic?

Homework Statement ##\sum _{n=1}^{\infty }\left[ \left( -1\right) ^{n}\right] \dfrac {\sqrt {n}} {1+2\sqrt {n}}##Homework Equations Alternating Series test, Absolute convergence theorem, p-series, and test for divergence. The Attempt at a Solution The alternating series test tells us that the...

Homework Statement Hello, I have a question concerning convergence of the non-monotonic sequences which takes place when the Cauchy criterion is satisfied. I understand that |a_n - a_m| <ε for all n,mN\ni Homework Equations What I don't see is how (a_{n+1} - a_n) →0is not...

I just need to know how you determine if a series of convergent or divergent. I have this example in which I know is divergent I just don't know why: summation (n=1 to infinity) 1/(2n) The first couple of terms are 1/2 + 1/4 + 1/6 + 1/8 + ... Up until that point, it's already beyond...

Homework Statement There are 3 parts to this problem: (a) \; \sum^{\infty}_{n=1} \frac{n^4}{4^n} (b) \; \sum^{\infty}_{n=1} \left( \frac{n+8}{n} \right)^n (c) \; \sum^{\infty}_{n=1} \frac{5^n-8}{4^n+11} The attempt at a solution (a) I've used the Ratio test. So, u_n=\frac{n^4}{4^n} and...

Homework Statement ##\sum _{n=1}ne^{-n}## Homework Equations Ratio Test Integral Test The Attempt at a Solution I know that by the ratio test, it converges absolutely. But, I am unable to determine its convergence through the integral test . Could someone help? I thought that the...

Homework Statement I'm given this series and asked whether it converges, absolutely converges, or diverges. Ʃ(n=0 to infinity) [2(-1^n)(3^(n+1))]/5^n Homework Equations The Attempt at a Solution The answer states that it is absolutely convergent, and that it converges to 15/4. Everything...

Homework Statement ##\sum \dfrac {1+2^{n}} {3^{n}}## According to Wolfram Alpha the sum is 5/2. But, I think that my method is fine and shows another result. The Attempt at a Solution ##\sum \dfrac {1+2^{n}} {3^{n}}=\sum \left[ \left( \dfrac {1} {3}\right) ^{n}+\left( \dfrac {2} {3}\right)...

Convergence for a series: what is wrong with my method?? Homework Statement For the following series, write formulas for the sequence an, Sn, and Rn, and find the limits of the sequences as n→∞. Homework Equations Sn is the partial sum of the series. Rn is the remainder and is...

Homework Statement Wondering if you guys could check my proof. This is my first problem with sequences of functions Let a > 0 and f_{n}(x) = \frac{nx}{1+nx}. a) Show that the sequence of functions (f_{n}) converges pointwise on [0,∞) b) Show that (f_{n}) converges uniformly on [a,∞)...

Radius of convergence of the series n^2(x^n)/(3n!) I am stumped the question is: find the radius and interval of convergence of the following series {sum_(n=1)^(Infinity)}((n^2)(x^n))/(3*6*9***3n) I'm assuming that equal to ((n^2)(x^n))/(3n)! then lim_(n->infinity) of...

Homework Statement A sequence {an} defined recursively by a1=1 and an+1=\frac{1}{2+a subn}, n\geq1. Show that the sequence is convergent. Homework Equations If a sequence is bdd below and decreasing or it is bdd above and increasing, then it is convergent. The Attempt at a Solution...

Dear friends, \sum_{x=1}^{\infty}\frac{1}{x} diverges. But \sum_{x=1}^{\infty}\frac{1}{x^{2}}=\frac{\pi^{2}}{6} How can we prove that \sum_{x=1}^{\infty}\left(\frac{1}{x^{\left(1+epsilon\right)}}\right) converges to a finite value? Thanks in advance. Bincy.

Homework Statement What is the radius of convergence of the Taylor Series of the function f(z) = z cot(z), at the point z = 0? Homework Equations Taylor series is given by: \sum_{k=0}^{\infty} \frac{f^{(k)}(z_{0})}{k!} (z - z_{0}) And the radius R by: \lim_{n \to \infty}...

State whether the sequence converges as n--> ##∞##, if it does find the limit i'm having trouble with these two: n!/2n and ∫ e-x2 dx now I know they're special forms so the ordinary tricks won't work. Any help or hints?

Consider a sequence \{ a_{n} \} . If \lim_{n→∞}a_{n} = L Prove that \lim_{n→∞}a_{n-1} = L I am trying to use the Cauchy definition of a limit, but don't know where to begin. Thanks. BiP

Homework Statement Find the interval of convergence for the given power series. Sum from n=1 to infinty of (x-11)^n / (n(-9)^n) Homework Equations The Attempt at a Solution I used the ratio test and I'm getting 2<x<20, but that doesn't seem to be right. I get abs(1/9*(x-11)) <...

\sum (1-\frac{1}{r})^{r^2} Does this converge or diverge.(r=1..inf) I have tried the following but do not think it is adequate(or correct for that matter) (1-\frac{1}{r})^r (1-\frac{1}{r})^r = (1-\frac{1}{r})^{r^2} and lim (1-\frac{1}{r})^r -> \frac{1}{e} thats given from a...

Prove that: (1-\frac{1}{n})^n \rightarrow \frac{1}{e} as n \to \infty you may use the fact that (1+\frac{1}{n})^n \rightarrow e I have no idea where to even begin, can someone point me in the right direction ?

Homework Statement \sum_{n=0}^{\infty}\frac{1}{n^2+3n+2} The attempt at a solution I'm wondering if there is only one way of solving this? Here is what I've done: First, converting into partial fractions. Is there a way to do it without converting to partial fractions...

Homework Statement I'm not sure how to do the notation on here but. Does this series converge or diverge. If it converges, then to what value. The series: Sum from 1 to infinity of [(-1)^n * n / (n^2-4n-4)] Homework Equations It tells me to use the ratio test The Attempt at a...

Homework Statement Is the sequence {n/(n^2+1)} convergent, and if so, what is it's limit?Homework Equations The Attempt at a Solution I believe it does converge because the higher power is in the denominator, so thus, it's limit is 0. Any help or hints on if I'm headed in the right direction...

Determine the values of "r" for which rn converges. Is there a specific procedure I should try to apply to figure this out? The only things I could intuitively come up with that will converge in this scenario are when -1 ≤ r ≤ 1...is there anything else to this?

I am trying to understand the idea of annulus of convergence. This is the example I have been looking at but it has me completely stumped. [∞]\sum[/n=1] (z^n!)(1-sin(1/2n))^(n+1)! + [∞]\sum[/n=1] (2n)!/[((n!)^2)(z^3n)] All of the examples I have worked on in the past have been...

State whether the sequence converges and if so, find the limit (n+1)1/2/2(n)1/2 ok so I got that it converges to 1/2, my question more so lies in the fact that why are we able to factor out a (n)1/2 from the term in the numerator? Isn't it only the denominator that we are concerned about...

Show that the infinite product f(z) = \prod\limits_{n = 0}^{\infty}(1 + z^{2^n}) converges on the open disc D(0,1) to the function 1/(1 - z). Is this convergence uniform on compact subsets of the disc? This should actually be done by the comparison test. For |z| < 1, we have that $$...

Given a totally finite measure μ defined on a \sigma-field X, define the (pseudo)metric d(A,B)=μ(A-B)+μ(B-A), (the symmetric difference metric), it can be shown this is a valid pseudo-metric and therefore the metric space (X',d) is well defined if equivalent classes of sets [A_\alpha] where...

$f(z) = \prod\limits_{n=1}^{\infty}\left(1+z^{2^n}\right)$ converges on the open disc $D(0,1)$ to the function $\dfrac{1}{1-z}$. To show convergence, I look at $$ \sum_{n=1}^{\infty}\left|z^{2^n}\right| $$ correct?The sum, $\sum\limits_{n = 0}^{\infty}|z|^{2^{n}}$, converges for $|z| < 1$ i.e...

Homework Statement Homework Equations Monotone Convergence Theorem: http://img696.imageshack.us/img696/5469/mct.png The Attempt at a Solution I know this almost follows from the theorem. But I first need to write \displaystyle \int_{I_n} f = \int_S f_n for some f_n in such a...

I'm having a lot of trouble with the subject. Here's one example I'd like explained. F(t_1, t_2) = \int \limits_0^1 x^{t_1}\ln^{t_2}\frac{1}{x} dx The book asks to find for what \vec{t} F converges. The answer is \vec{t}\in(-1; \infty)^2, but I don't see how to get that. In general, what...

Homework Statement I would just like to be pointed in the right direction. I have this theorem: Let E be a measurable set of finite measure, and <fn> a sequence of measurable functions that converge to a real-valued function f a.e. on E. Then given ε>0 and \delta>0, there is a set...