Convergence Definition and 1000 Threads

Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...

Homework Statement \sum_{n=2}^{\infty}sin(\frac{\pi*n}{2})/{2}I don't have a solution, and wondered if the execution is correct. The Attempt at a Solution I thought that one can use comparison test where; \sum b_n= \frac{1}{n^{1/2}}. Since p<1 ---> divergent. But many of the students says it...

Hey! :o I want to find for the following series the radius of convergence and the set of $x\in \mathbb{R}$ in which the series converges. $\displaystyle{\sum_{n=0}^{\infty}\frac{n}{2^n}x^{n^2}}$ $\displaystyle{\sum_{n=0}^{\infty}\frac{1}{(4+(-1)^n)^{3n}}(x-1)^{3n}}$ I have done the...

Homework Statement ##I## a set of real number and ##f_k : I \rightarrow \mathbb{R}## a succession of real functions defined in ##I##. We say that ##f_k## converges punctually in ##I## to the function ##f : I \rightarrow \mathbb{R}## if $$\lim_{k \to \infty} f_k(x) = f(x), \hspace{1cm} \forall x...

I have $$\sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$$ I'm trying the limit comparison test, so I let $$ b = \frac{1}{n^{\frac{9}{8}}}$$ and $a = \sum_{n = 2}^{\infty} \frac{(lnn)^ {12}}{n^{\frac{9}{8}}}$ $\frac{a}{b} = (lnn)^ {12}$ therefore I know the limit of this as n...

It seems to me that convergence rounds away the possibility of there being a smallest constituent part of reality. For instance, adding 1/2 + 1/4 + 1/8 . . . etc. would never become 1, since there would always be an infinitely small fraction that made the second half unreachable relative to the...

I have a set of data that I've been working with that seems to be defined by the sum of a set of exponential functions of the form (1-e^{\frac{-t}{\tau}}). I've come up with the following series which is the product of a decay function and an exponential with an increasing time constant. If this...

Homework Statement The question was to find the interval of convergence for a series. Homework Equations an+1/an The Attempt at a Solution

In Mary L. Boas' Mathematical Methods in the Physical Science, 3rd ed, on page 17 it goes over absolute convergence, and defines the test for alternating series as follows: An alternating series converges if the absolute value of the terms decreases steadily to zero, that is, if |an+1| ≤ |an|...

$\text{Evaluate answer from }\textit{ TI-Nspire CX CAS}$ \begin{align*} \displaystyle S_k&=\sum_{k=1}^{\infty} \left[\frac{(-2)}{9^{k+1}}\right] =\frac{-2}{99} \\ \end{align*} ok wasn't sure what weapon of choice to use $\tiny{206.10.3.75}$ ☕

http://‪C:\Users\johny\Downloads\q4.jpg 1. Homework Statement Hi, so the question is I have to tell if this integral diverges or converges.(without solving it) integral(1/(e^x sqrt(x)))dx from 1 to +inf Homework Equations integration techniques. The Attempt at a Solution my answer: let 1/e^x...

Prove that if a subsequence of a Cauchy sequence converges then so does the original Cauchy sequence. I'm assuming that we're not allowed to use the fact that every Cauchy sequence converges. Here's my attempt: Let $\displaystyle\{s_n\}$ be the original Cauchy sequence. Let $\displaystyle...

It is perfectly fine to do the following: ##\displaystyle{\int_{-\infty}^{\infty}\ d\phi\ e^{-\phi^{2}/2}e^{-\lambda \phi^{4}/4!} = \int_{-\infty}^{\infty}e^{-\phi^{2}/2}\sum\limits_{n=0}^{\infty}}\frac{(-\lambda\phi^{4})^{n}}{(4!)^{n}\...

I'm using the book of Jerome Keisler: Elementary calculus an infinitesimal approach. I have trouble understanding the proof of the following theorem. I'm not sure what it means. Theorem: "An increasing sequence <Sn> either converges or diverges to infinity." Proof: Let T be the set of all real...

Homework Statement The problem states: In the harmonic series ##\sum_{1}^{\infty} \frac{1}{k}##, all terms for which the integer ##k## contains the digit 9 are deleted. Show that the resulting series is convergent. Hint: Show that the number of terms ##\frac{1}{k}## for which ##k## contains no...

Hi guys, I am doing this question of alternating series test. And I was following the below principles when solving the problem. Sorry I don't know how to type in the math language. I got 4, 8, 9, 10 as the answers. But the system rejected this without any explanation. Can someone throw a...

Hi everybody! I'm preparing an exam of "Analysis II" (that's how the subject's called in German), and I have trouble understanding how to find the limit of a multivariable function, especially when it comes to proving the uniform convergence. Here is an example given in the script of my teacher...

Let ##\boldsymbol{l}:\mathbb{R}\to\mathbb{R}^3## be the piecewise smooth parametrization of an infinitely long curve ##\gamma\subset\mathbb{R}^3##. Let us define $$\boldsymbol{B}(\boldsymbol{x})=\frac{\mu_0...

So I have $$\sum_{n = 2}^{\infty} \frac{1}{nln(n)}$$ I'm trying to apply the limit comparison test, so I can compare it to $b_n$ or $\frac{1}{n}$ and I can let $a_n = \frac{1}{nln(n)}$ Then I get $$\lim_{{n}\to{\infty}} \frac{n}{nln(n)}$$ Or $$\lim_{{n}\to{\infty}} \frac{1}{ln(n)}$$ Which is...

Supposing I have this expression: $$\sum_{n = 1}^{\infty} \frac{x^n}{3^n}$$ and I need to find the values for x for which this converges and the radius of convergence. I can use the radius test: $$\lim_{{n}\to{\infty}} |\frac{{x}^{(n + 1)} 3^n}{{3}^{(n + 1)} x^n}|$$ and this equals...

I have this series: $$\sum_{k = 1}^{\infty} {4}^{\frac{1}{k}}$$ To solve this, I am trying to compare it to this series $$\sum_{k = 1}^{\infty} {4}^{k}$$ So, I can let $a_k = {4}^{\frac{1}{k}} $ and $b_k = {4}^{k}$ These seem to be both positive series and $ 0 \le a_k \le b_k$ Therefore...

I have $$\sum_{n = 2}^{\infty} \frac{{(\ln\left({n}\right)})^{12}}{n^{\frac{9}{8}}}$$ We can compare it to $ \frac{1}{{n}^{\frac{1}{8}}}$. $ \sum_{n = 1}^{\infty} \frac{1}{{n}^{\frac{1}{8}}}$ diverges because $p < 1$ in this case. So, if I can prove that $...

Homework Statement determine whether the series below converges. ##\sum_{n=1}^\infty 2^n.n+1,√(n^4+4^n.n^3)## Homework EquationsThe Attempt at a Solution

Hello, I have two questions regarding the Radius of convergence. 1. What should we do at the interval (R-eps, R) 2. It used definition to prove radius of convergence, but I am not sure if it is necessary-sufficient condition of convergence. I get that this can be a sufficient condition but not...

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 can't seem to get around it so here it goes ...

Hello, I am want to prove that: $$ \sum_{1}^{\infty} \frac{1}{n^{2} + 1} < \frac{1}{2} + \frac{1}{4}\pi $$ Cauchy's Convergence Integral If a function decreases as n tends to get large, say f(x), we can obtain decreasing functions of x, such that: $$ f(\nu - 1) \geqslant f(x) \geqslant...

I'm trying to understand this python CAMB code: http://camb.readthedocs.io/en/latest/CAMBdemo.html Scroll down to In[29] and In[30] to see it. It's an integration over chi (comoving distance), yet scipy.integrate.quad is not called. It seems that the fun stuff happens in the last for-loop in...

Find the values of ##x## for which the following series is convergent. I compared the series with the harmonic series and deduced it is always divergent. I used ##y^p<e^y## for large ##y##. I used a different method from the answer given, which I don't understand. When ##k=1##...

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...

Consider a sequence with the ##n^{th}## term ##u_n##. Let ##S_{2m}## be the sum of the ##2m## terms starting from ##u_N## for some ##N\geq1##. If ##\lim_{N\rightarrow\infty}S_{2m}=0## for all ##m##, then the series converges. Why? This is not explained in the following proof:

Homework Statement From the given ans , i knew that it's conditionally convergent (by using alternating test) i can understand the working to show that it's conditionally convergent . But , i also want to show it as not absolutely convergent ... Homework EquationsThe Attempt at a Solution...

Does the sequence \{f_n\}=\{\cos{(2nt)}\} converge or diverge in Banach space C(-1,1) endowed with the sup-norm ||f||_{\infty} = \text{sup}_{t\in (-1,1)}|f(t)| ? At first glance my intuition is that this sequence should diverge because cosine is a period function. But how to really prove...

Homework Statement How can I show that this series is convergent for z=1 and z<1 and divergent for z>1 $$\sum _{p=1}^{\infty }\dfrac {z^{p}} {p^{3/2}}$$ Homework Equations http://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx The Attempt at a Solution Using the ratio test I've...

Hello, I am struggling to understand a simple question on limits. I have watched a video trying to explain the theory and even have the answer right in front of me but I still don't understand. Could somebody please explain the steps in detail for me just for the first question as I'm hoping...

I'm trying to find the answer to a question similar to this posted it earlier but in the wrong section I think and not explained well. $$ \sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n \ \ \rm ?$$ Find the interval of convergence? I tried using the root test and got from 0 to 1 but when I...

Homework Statement Let f(x)= (1+x)4/3 - In this question we are studying the Taylor series for f(x) about x=2. This assignment begins by having us find the first 6 terms in this Taylor series. For time, I will omit them; however, let's note that as we continuously take the derivative of this...

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >[/color] How would you find the radius of convergence for the taylor expansion of: \begin{equation} f(z)=\frac{e^z}{(z-1)(z+1)(z-3)(z-2)} \end{equation} I was thinking that you would just...

The criteria for testing for convergence with the alternating series test, according to my book, is: Σ(-1)n-1bn With bn>0, bn+1 ≤ bn for all n, and lim n→∞bn = 0. My question is about the criteria. I'm running into several homework problem where bn is not always greater than bn+1, such as the...

Homework Statement Find all values of x such that the given series would converge Σ6n(x-5)n(n+1)/(n+11) Homework EquationsThe Attempt at a Solution By doing the ratio test I found that lim 6n(x-5)n(n+1)/(n+11) * (n+12)/[6n+1(x-5)n+1(n+2)] n→inf equals 1/(6(x-5)) * lim...

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...

Hello; I'm struggling with pointwise and uniform convergence, I think that examples are going to help me understand Homework Statement Consider the Fourier sine series of each of the following functions. In this exercise de not compute the coefficients but use the general convergence theorems...

Let f(x) = (1+x)-4 Find the Taylor Series of f centered at x=1 and its interval of convergence. \sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!} is general Taylor series form My attempt I found the first 4 derivatives of f(x) and their values at fn(1). Yet from here I do not know how to find the...

Does there exist a proof of the following: It is well known that Picard successive approximations on the Fredholm-equation (1) $y(x)=f(x)+{\lambda}_{1}\int_{a}^{b} \,k(x,s)y(s)ds$ written in operator form as $y=f+{\lambda}_{1} Ky$ converges if (2) $|{\lambda}_{1}|. ||K||<1$ where $K$...

Homework Statement Hi everybody! I'm a little struggling to fully understand the idea of radius of convergence of a function, can somebody help me a little? Are some examples I found in old exams at my university: Calculate the radius of convergence of the following power series: a)...

Homework Statement \sum\limits_{n=1}^{\infty}\frac{n-1}{(n+2)(n+3)} Homework Equations S=\sum\limits_{n=1}^{\infty}a_n (1) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\gt 1\rightarrow S\ is\ divergent (2) \lim\limits_{n\rightarrow\infty}\frac{a_{n+1}}{a_n}\lt 1\rightarrow S\ is\...

Homework Statement a) If ##f: [0,1] \rightarrow \mathbb{R}## is continuous and ##\int^{b}_{a} f(x)dx = 0## for every interval ##[a,b] \subset [0,1]##, then ##f(x)=0 \forall x \in [0,1]## b) Let ##f: [0,\infty) \rightarrow [0,\infty)## be continous. If ##\int^{\infty}_{0} f(x)dx## converges...

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...

Here's what I'm confused about: The harmonic series 1/n is divergent (because you're just infinitely adding numbers) but the series (1/2)^(n+1) is convergent. This doesn't make sense to me because by plugging in larger and larger numbers, you are still adding small numbers infinitely which...

Which of these converge? 1. 5n/ (2n-1) 2. e^n/ n 3. e^n/ (1+e^n) Attempt: 1) lim n-> ∞ 5n/(2n-1) = 5n/2n = 5/2 ≠ 0 so diverge? 2) Change n to x e^x/ (1 + e^x) Antiderivative: ln |1 + e^x| lim t->∞ of ln |1 + e^x| ln |1 + e^∞| - ln |1 + e^0| Got stuck here 3) Help :(

n^2 - 1 / (n^3 + 6n) If I use the nth divergence test, I plug ∞ in (limit as n -> ∞) for n and since the degree on the bottom is larger I get 0, which means it converges. However, if I use the limit comparison test and compare it to: n^2/n^3, which = 1/n, which diverges -> n^2 - 1 / (n^3 + 6n)...