Convergence Definition and 1000 Threads

I am attempting to evaluate the radius of convergence for a series that goes from k=0 to infinity. The series is given by (k*x^k)/(3^k). I have begun by using the ratio test and have gotten to the point L = (k+1)*x/3k Now i know i can find out the radius of convergence by simply saying R =...

Hello! (Wave) $$e^x= \sum_{n=0}^{\infty} \frac{x^n}{n!} \forall x \in \mathbb{R}$$ i.e. the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$. Could you explain me how we deduce that the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$? Do...

##a_1=1## ##a_{n+1}=\frac{1}{2} ( a_{n} + \frac{b}{a_n} )## This should converge to ##\sqrt{b}## but I seem not to be able to prove this. Could someone give me a hint.

Homework Statement "Determine whether the following series converge or diverge. If the series is geometric or telescoping, find its sum.": ## \left ( \sum_{k=1}^\infty2^{3k} *3^{1-2k} \right)## Homework Equations [/B] The different tests for convergence? The Attempt at a Solution Ok...

Homework Statement Consider ##\sum\limits_{n=0}^{\infty} \frac{n+1}{(2n)!}(x+1)^{2n+1}##. Find the interval of convergence and sum of the power series. Homework EquationsThe Attempt at a Solution According to the textbook: given the power series ##\sum a_n(x-c)^n## the radius of convergence...

Hey everyone, I'm currently in Calc 2 and the only thing I seem to be having a problem with is a couple of the convergence tests. When I take pretty much any math course, I always use mathematica to help check my answers when I'm doing HW or practicing so I don't waste time. My question is...

I have the integral 1/(x^0.25 - 2) dx between 500 to 16, and am trying to find whether it converges or diverges. I have sketched the graph and noticed that their is an asymptote at x=16 (hence why the integral is improper for these boundaries). I am now trying to evaluate the limits to see if...

Homework Statement I need to show that \sum\limits_{n=0}^\infty \frac{sin^{4}(\frac{n\pi}{4})}{n^2} = \frac{\pi^{2}}{16} Homework Equations I have this property for odd n \sum\limits_{n=0}^\infty \frac{1}{n^2} = \frac{\pi^{2}}{8} The Attempt at a Solution [/B] I have no idea how to do...

How can I find out if 1/n! is divergent or convergent? I cannot solve it using integral test because the expression contains a factorial. I also tried solving it using Divergence test. The limit of 1/n! as n approaches infinity is zero. So it follows that no information can be obtained using...

Homework Statement ∞ ∑ tan(1/k) k=5 show that it is convergent or divergent Homework EquationsThe Attempt at a Solution i used ratio test, but it's equal to 1, it means no works... i used divergence test, it equals to 0, no work too... so what should i do? i don't know how to use...

Homework Statement Determine whether the series is converging or diverging Homework Equations ∞ ∑ 1 / (3n +cos2(n)) n=1The Attempt at a Solution I used The Comparison Test, I'm just not sure I'm right. Here's what I've got: The dominant term in the denominator is is 3n and cos2(n)...

Hello, For the exercises in my textbook the directions state: "Use power series operations to find the Taylor series at x=0 for the functions..." But now I'm confused; when I see "power series" I think of functions that have x somewhere in them AND there is also the presence of an n. Here...

Homework Statement Does (1/(k!)) converge? Homework Equations [/B] Convergence Tests?The Attempt at a Solution I thought I could just simply use the divergence test, but I'm not sure if that only tells you if it's divergent and not whether it is convergent or not. lim(k>inf) (1/(k!)) = 0...

Homework Statement [/B] This is the question I have (from a worksheet that is a practice for a quiz). Its a conceptual question (I guess). I understand how to solve ratio test problems. "Is this test only sufficient, or is it an exact criterion for convergence?" Homework Equations Recall the...

Hi, i solved a model but i got an error saying that "The solver engine was unable to converge on a solution for the non linear problem as constrained". Can someone offer advice on how i can fix this error?

Investigate the convergence of the integral $\int_{0}^{\infty} \frac{x^{ \frac{2}{3}} + \frac{2}{3} \ln x }{1+x^2} \mbox{d}x$

Investigate the convergence of the integral $\int_{0}^{\infty} \frac{-\ln \left[ \frac{1}{a} x ^{ \frac{2}{a-1} } e ^{- x^{\frac{2}{a} }} \right] }{1+x^2} \mbox{d}x$ for $a \ge 2$

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...

Homework Statement For ##|z-a|<r## let ##f(z)=\sum_{n=0}^{\infty}a_n (z-a)^n##. Let ##g(z)=\sum_{n=0}^{\infty}b_n(z-a)^n##. Assume ##g(z)## is nonzero for ##|z-a|<r##. Then ##b_0## is not zero. Define ##c_0=a_0/b_0## and, inductively for ##n>0##, define $$ c_n=(a_n - \sum_{j=0}^{n-1} c_j...

I have a question pertaining to a computation I'm trying to carry out. Without getting too much into the details, I have a finite integral over two variables. One integral vanishes, and one diverges allowing for the finite value. I had to regulate the divergent integral so I introduced a...

Homework Statement \ell is the set of sequences of real numbers where only a finite number of terms is non-zero, and the distance metric is d(x,y) = sup|x_n - y_n|, for all n in naural-numbers then the sequence u_k = {1,\frac{1}{2},\frac{1}{3},...,\frac{1}{k}, 0,0,0...} and...

Homework Statement Does sum from n=1 to n=infinity of 1/[n^(1+1/n)] converge or diverge. Homework Equations ^^^^^^^^^^^^^^^ The Attempt at a Solution The general term goes to 0 and its a p-series with p>1, but for large n the series becomes 1/n pretty much so, even tho p>1 is it divergent?

Homework Statement Does \frac{2^{n}}{n!} converge or diverge? The Attempt at a Solution Is there more than one way to prove this? I would appreciate a few directions. I've been trying the Squeeze theorem for a long time. I said 1/n! was smaller, but I have no damn idea how to say what's...

Homework Statement For which of them will the corresponding fixed point iteration xk+1 = g(xk) be locally convergent to the solution xbar in [0, 1]? (The condition to check is whether |g'(xbar)| < 1.) A) 1/x2 -1 B)... C)... compute xbar to within absolute error 10-4. Homework Equations 3. The...

Homework Statement z ∈ ℂ What is the radius of convergence of (n=0 to ∞) Σ anzn? Homework Equations I used the Cauchy-Hardamard Theorem and found the lim sup of the convergent subsequences. a_n = \frac{n+(-1)^n}{n^2} limn→∞ |an|1/n The Attempt at a Solution I think that the radius of...

Homework Statement Part a.) For a>0 Determine Limn→∞(a1/n-1) Part b.) Now assume a>1 Establish that Σn=1∞(a1/n-1) converges if and only if Σn=1∞(e1/n-1) converges. Part c.) Determine by means of the integral test whether Σn=1∞(e1/n-1) converges Homework Equations Integral Test Limit...

Homework Statement Find whether the integral is convergent or not, and evaluate if convergent. Homework Equations integral 1/sqrt(x^4+x^2+1) from 1 to infinity The Attempt at a Solution 1/sqrt(x^4+x^2+1)<1/sqrt(x^4) 1/sqrt(x^4)=1/x^2 which is convergent for 1 to infinity and is 1 therefore...

1. After finding out that the wave function ##\Psi(z) \sim Ae^{\frac{-z^{2}}{2}}## in the limit of plus or minus infinity Griffiths separates the function into two parts ##\Psi(z)=h(z)e^{\frac{-z^{2}}{2}}## My question will be about a certain aspect of the function ##h(z)## After solving the...

Homework Statement So my question was Sum- (n=2) ln(n)/n Homework Equations I noticed that you can only limit comparison, because so far, I have tried doing all the other test such as the nth term test, p-series, integral(i have no idea how to integrate that). The Attempt at a Solution

So I'm reading "An Introduction to Wavelet Analysis" by David F. Walnut and it's saying that the following sequence " (x^n)_{n\in \mathbb{N}} converges uniformly to zero on [-\alpha, \alpha] for all 0 < \alpha < 1 but does not converge uniformly to zero on (-1, 1) " My problem is that isn't...

I know when a series is conditionally convergent and I understand that being conditionally convergent means that rearrangement of the terms will not always lead to the same sum, but I am unsure why exactly this is important? I have not really experienced a time where I am rearranging terms...

Hi all, I am analysing a 2D D-shaped Neo Hookean model in contact with a Rigid link. The details are in the input file attached. Can someone guide me on how to solve this issue?:headbang: Thanks, Bruce

Homework Statement I ultimately want to discuss convergence of the integral \int_{0}^{\infty}\frac{1}{\sqrt{x}e^{\sqrt{x}}}dx[/B]Homework Equations \int_{c}^{\infty}\frac{dx}{x^{p}} is convergent near x approaching infinity for p>1 3. The Attempt at a Solution While I understand that the...

Homework Statement I have been asked to prove the convergence or otherwise of ∑∞n=1 n/(3n + n2). In the example solution, with the aim to prove divergence by comparison with the Harmonic Series, the lecturer has stated that n/(3n + n2) ≥ n/(4n2) = 1/4n and which diverges to +∞. I was...

Homework Statement [/B] I have to find the radius of convergence and convergence interval. So for what x's the series converge. The answer is supposed to be for every real number. So the interval is: (-∞, ∞). So that must mean that the limit L = 0. So the radius of convergence [ which is...

Homework Statement Prove the following double integral is convergent. ##\int_0^1 \int_0^1 \frac{1}{1-xy}\, dx \, dy## The Attempt at a Solution This was a bonus question on my final exam in calc 3 yesterday, I just want to show my steps and see if they were right. So I realized that...

Homework Statement Consider the series: \sum\limits_{k=17}^\infty (-1)^{k}(\sqrt{k-3}-\sqrt{k-5}) Homework EquationsThe Attempt at a Solution First, I will attempt to determine whether it is absolutely convergent: \lim\limits_{k\to\infty} \left(\sqrt{k-3}-\sqrt{k-5}\right) = 0 Since the limit...

Homework Statement Find L[x(t)], where $$ x(t) = tu(t) + 3e^{-1}u(-t) $$ Also determine the region of convergenceHomework EquationsLaplace properties, Laplace table: L[te-at = 1/(s+a)2 L[u(t)] = 1/s L[t] = 1/s2 The Attempt at a Solution I don't really know what to do with this as my table...

Homework Statement ∞ n=3 ∑ ((-1)n (x+3)3n)/(2nlnn) Find radius of convergence, interval of convergence, values for x which series is: absolutely convergent, conditionally converge or divergence. Homework EquationsThe Attempt at a Solution I applied the Ratio Test and got |(x+3)3| lim...

Hey! :o If $f_n, f \in L^p, 1\leq p < +\infty$ and $f_n \rightarrow f$ almost everywhere, and $||f_n||_p \rightarrow ||f||_p$, then $f_n\rightarrow f$ as for the norm. Could you give me some hints how to show it?? (Wondering) What does convergence as for the norm mean?? (Wondering)

Homework Statement ∑ x2n / n! The limits of the sum go from n = 0 to n = infinity Homework EquationsThe Attempt at a Solution So I take the limit as n approaches infinity of aa+1 / an. So that gives me: ((x2n+2) * (n!)) / ((x2n) * (n + 1)!) Canceling everything out gives me x2 / (n + 1)...

Hi, I am likely just missing something fundamental here, but I recently just revisited series and am looking over some notes. In my notes, I have written that if ## \lim_{x \to +\infty} \frac{a_{n+1}}{a_n} = L ## Then ## | x - x_o | = 1/L ## But shouldn't the correct expression be $$ | x -...

Hello. In my complex analysis book I've read a theorem which says that if a sequence $$\{ f_n \}$$ of holomorphic functions on a domain $$\Omega$$ converges pointwise to a function $$f$$, then $$f $$ is holomorphic on a dense, open subset of $$\Omega$$. I know how to prove this theorem. I...

Find the LT and specify ROC of: x(t) = e-at, 0 ≤ t ≤ T = 0, elsewhere where a > 0 Attempt: X(s) = - 1/(s+a)*e-(s+a) integrated from 0 to T => -1/(s+a)[e-(s+a) + 1] Converges to X(s) = 1/(s+a) , a ⊂ R, if Re{s} > -a for 0≤t≤T Elsewhere ROC is empty (LT doesn't exist). Is this...

Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...

Homework Statement an = [sin(n)]/n Prove that this sequence converges using Cauchy theorem Homework Equations Cauchy theorem states that: A sequence is called a Cauchy theorem if for all ε > 0, there exists N , for all n > N s.t. |xn+1 - xn| < εI do not know how to approach this proof. I...

Hello people, In case if I am typing the question in the wrong forum please redirect me. OK, here it goes... I have this stupid question: Suppose we have a sequence 1, 1, 1, 1, 1, 1... It converges to '1'. Consider 1, 0, 1, 0, 1, 0... it diverges right? What about a sequence 1, 1, 0, 1, 1...

Homework Statement Suppose (## a_n ##) is a sequence and let l\in\mathbb R. Let us say that (## a_n ##) is "super convergent" to ##l## if there exists N\in\mathbb N such that for every ε>0 we have ##n \geq N## ⇒ |(## a_n - l##|<ε . Show that if (## a_n ##) super converges to l then (## a_n...

B]1. Homework Statement [/B] Find whether the series is convergent or divergent Homework Equations The Attempt at a Solution By ratio test I have, I would apply L'Hôpital's rule to find the value of limit but before that how do i simplify the expression? It has fractional part both in the...

Homework Statement Let V consist of all infinite sequences {xn} of real numbers for which the series summation xn2 converges. If x = {xn} and y = {yn} are two elements of V, define (x,y) = summation (n=1 to infinity) xnyn. Prove that this series converges absolutely. Homework Equations The...