# What is Convergence: Definition and 1000 Discussions

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.

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1. ### Ratio Test vs AST

Hi, I'm having difficulty understanding why the interval of convergence is (0, 18]. When I tested x=18, I got the following conclusion using the ratio test. When I attempt using AST, the function still diverges as the lim (n -> inf) = 2^n / n ≠ 0. What am I missing? Thanks!
2. ### Valid conclusion for an absolutely convergent sequence

Hello, this is my attempt for #19 for 11.6 of Stewart's “Multivariable Calculus”. The question is to determine whether the series is absolutely convergent, conditionally convergent, or divergent. The answer solutions used a ratio test to reach the same conclusion but I used the comparison test...
3. ### A "spiral" in the Complex plane

I understand that the "spiral" converges to 1+i-1/2-i/3!+1/4!+i/5!-1/6!-i/7!... . It splits into two: one for Re, 1-1/2+1/4!-1/6!..., and the other for Im, 1-1/3!+1/5!-1/7!... . Any hints on how to compute them?
4. ### Proving convergence of rational sequence

For this problem, The solution is, However, does someone please know why this did not use ##2n ≤ 2n^2 + 2n + 1## which would give ##\frac{3n - 1}{2n^2 + 2n + 1} ≤ \frac{3n}{2n} = \frac{3}{2}##? In general, after solving many problems, it seems that when proving the convergence of a rational...

15. ### Determine Convergence/Divergence of Sequence: f(x)=ln(x)^2/x

##a_n= \left[\dfrac {\ln (n)^2}{n}\right]## We may consider a function of a real variable. This is my approach; ##f(x) =\left[\dfrac {\ln (x)^2}{x}\right]## Applying L'Hopital's rule we shall have; ##\displaystyle\lim_ {x\to\infty} \left[\dfrac {\ln (x)^2}{x}\right]=\lim_ {x\to\infty}\left[...
16. ### POTW Convergence in Probability

Prove that if ##\{X_n\}_{n = 1}^\infty## is a sequence of real random variables on probability space ##(\Omega, \mathscr{F},\mathbb{P})## such that ##\lim_n \mathbb{E}[X_n] = \mu## and ##\lim_n \operatorname{Var}[X_n] = 0##, then ##X_n## converges to ##\mu## in probability.
17. ### Convergence of a Series: Radius and Endpoints

Greetings According to my understanding: if x converges in 4 means that the series converges -1<x+3<7 but the solution says C Any hint? thank you!
18. ### A Convergence issue in this Least Squares calculation

I'm computing the trajectory of a moving body and my net is composed by 5 stations. My observations are DTOA: difference in time of Arrival (they have been linearized). I am trying to use Least Squares with a linear model: Y = Ax + b, where Y are the observed measurements (DTOA), A the design...
19. ### Why are there clouds over the Intertropical Convergence Zone?

Intertropical Tropical Zone is the zone where north-east and south-east trade winds converge. This zone usually occurs over (I don’t know if “on” should be here) the equator. In the book The Atmosphere: An Introduction to Meteorology by Lutgens and Tarbuck (13th Edition), Figure 7.9 reads I...
20. ### I Proving a convergent sequence is bounded

Dear Everybody, I have a quick question about the \M\ in this proof: Suppose \b_n\ is in \\mathbb{R}\ such that \lim b_n=3\. Then, there is an \ N\in \mathbb{N}\ such that for all \n\geq\, we have \|b_n-3|<1\. Let M1=4 and note that for n\geq N, we have |b_n|=|b_n-3+3|\leq |b_n-3|+|3|<1+3=M1...
21. ### A Residual of PDEs as convergence criteria of numerical solution

Use a numerical method to solve a PDE f[u(x),u'(x),...]=0, where f is an operator, e.g. u'(x)+u(x)=0, and obtain a numerical solution v(x). Define f[v(x),v'(x),...] as the residual of the original PDE. Is this residual of the PDE widely used as the convergence criteria of the numerical solution...
22. ### The convergence Criteria ratio

Greetings all I have a question regarding the convergence criteria ratio, abs(an+1/an) or the n√abs(an) when the limit tend to a value less than 1 does it mean the serie is convergent or absolutely convergent? Thank you!
23. ### Question about the convergence of a series

Greetings! I have a question about one assumption regarding this question even though I agree with the answer but I have a doubt about A, because when we study the convergence of a serie we use the assymptotic approximation, so why A is not correct? thank you! when we
24. ### Bounded and monotonic sequences - Convergence

I would like some clarity on the highlighted part. My question is, consider the the attached example ##(c)##, This sequence converges ( by using L'Hopital's rule)...now my question is, the sequence is indicated on text as not being monotonic...very clear. Does it imply that if a sequence is not...
25. ### The radius of convergence of a series

Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!
26. ### Prove that the inner product converges

I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...
27. ### I Convergence of this Laplace transformation

I have a f(t) that is, e^(-t) *sin(t), now I calculate the Laplace transformation, that is: X(s) = 1 / ( 1 + ( 1 + s)^2 ) (excuse me but Latex seems not run ). Now I imagine the plane with Re(s), Im(s) and the magnitude of X(s). If i take Re(s) = -1 and Im(s) = 0, I believe I have X(s) = 1 ( s...
28. ### Is Convergence Possible from A to C?

If there are points A and B in space, If an object travels a distance from A to any other point between A and B, does that count as converging? And if the object from point A reach point C (a point between A and B) without traveling a distance, does that count as converging too? Like if the...
29. ### MHB Does This Sequence Converge Uniformly?

Hey! :giggle: We have the sequence of functions $$f_n=\sin (x)-\frac{nx}{1+n^2}$$ I want to check the pointwise andthe uniform convergence. We have that $$f^{\star}(x)=\lim_{n\rightarrow \infty}f_n(x)=\lim_{n\rightarrow \infty}\left (\sin (x)-\frac{nx}{1+n^2}\right )=\sin(x)$$ So $f_n(x)$...
30. ### MHB Extrema and convergence of sequence

Hey! :giggle: For $n\in \mathbb{N}$ let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ given by $f_n(x)=\frac{x+2n}{x^2+n}$. (a) Determine all (local and global) extrema of $f_n$ and the behaviour for $|x|\rightarrow \infty$. Make a sketch for $f_n$ and $f_n'$. Show that there exists $x_1<x_2<x_3<x_4$...
31. ### The convergence of a numerical series

Greetings here is the exercice My solution was as n^2+n+1/(n+1) tends asymptotically to n then the entire stuffs inside the sinus function tends to npi which make it asymptotically equal to sin(npi) which is equal to 0 and consequently the sequence is Absolutely convergent Here is the...
32. ### Finding the Radius of Convergence for Y=6x+16 - Troubleshooting and Solution

Greetings I have some problems finding the correct result My solution: I puted Y=6x+16 so now will try to find the raduis of convergence of Y so let's calculate the raduis criteria of convergence: We know that Y=6x+16 Conseqyently -21/6<=x<=-11/6 so the raduis must be 5/3. But this is not...
33. ### B Convergence of a sequence of averages of a convergent sequence

Let ##\epsilon>0##. Then there is an integer ##N>0## with the property that for any integer ##n\geq N##, ##|a_n-A|<\epsilon##, where ##A\in\mathbb{R}##. If for all positive integers ##n##, it is the case that ##|a_n-A|<\epsilon##, then the following must hold: \begin{eqnarray}...
34. ### Series Convergence: What Can the Nth Term Test Tell Us?

I'm not sure which test is the best to use, so I just start with a divergence test ##\lim_{n \to \infty} \frac {n+3}{\sqrt{5n^2+1}}## The +3 and +1 are negligible ##\lim_{n \to \infty} \frac {n}{\sqrt{5n^2}}## So now I have ##\infty / \infty##. So it's not conclusive. Trying ratio test...
35. ### Purcell EM Problem 1.2: Theory Behind Numerical Solution?

The author start of with $\frac{1}{(y+\sqrt{3})^2} - 2 \cdot \frac{1}{1 + y^2} \left( \frac{y}{\sqrt{1+y^2}} \right) = 0$ and arrives at the equation $y = \frac{(1+y^2)^{3/2}}{2(y+\sqrt{3})^2}$ The solution is merely by iterating (use an initial guess value of y, calculate the RHS, then use this...
36. ### I Fredholm's alternative & L2 convergence

Hello everyone, I'm currently going through Strauss "introduction to differential equations" and i can't get around a certain proof that he gives on chapter 11.5 page(327 (2nd edition)).Specifically, the proof refers to a certain version of Fredholm's alternative theorem. Assume that we are...
37. ### I Understanding the Laplace Transform of cos(t)/t

So, I know the direct definition of the Laplace Transform: $$\mathcal{L}\{f(t) \} = \int_0^\infty e^{-st}f(t)dt$$ So when I plug in: $$\frac{\cos(t)}{t}$$ I get a divergent integral. however:https://www.wolframalpha.com/input/?i=+Laplace+transform+cos%28t%29%2F%28t%29 is supposed to be the...
38. ### Finding the convergence of a parametric series

It is clear that the terms of the sequence tend to zero when n tends to infinity (for some α) but I cannot find a method that allows me to understand for which of them the sum converges. Neither the root criterion nor that of the relationship seem to work. I tried to replace ##\sqrt[n]{n}## with...
39. ### A problem with the convergence of a series

Good day I have a question about the convergence of the following serie I understand that the racine test shows that it an goes to 2/3 which makes it convergent but I also know that for a sequence to be convergent the term an should goes to 0 but the lim(n---->inf) ((2n+100)/(3n+1))^n)=lim...

43. ### MHB Fixed point iteration convergence

Question: For the following functions, does the fixed point iteration for finding the fixed point in $\left [ 0,1 \right ]$ converge for all first points $p_{0} \in \left [ 0,1 \right ]$? Justify your answer. a.$g(x) = e^{\frac{-x}{2}}$ b.$g(x) = 3x - 1$ Let me attempt for part a first...
44. ### Checking convergence of Gaussian integrals

a) First off, I computed the integral \begin{align*} Z(\lambda) &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}-\frac{\lambda}{4!}x^4\right) \\ &= \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} dx \exp\left( -\frac{x^2}{2!}\right) \exp\left(...
45. ### I Convergence criterion for Newton-Raphson

The Newton-Raphson algorithm is well-known: ##x_{n+1} = x_n - \frac{f(x_n)}{f'(x_{n})}## Looking at a few implementations online, I have encountered two methods for convergence: 1) The first method uses the function value of the last estimate itself, ##f(x_n)## or ##f(x_{n+1})##. Since at...
46. ### Problem in finding the radius of convergence of a series

Good day I'm trying to find the radius of this serie, and here is the solution I just have problem understanding why 2^(n/2) is little o of 3^(n/3) ? many thanks in advance Best regards!
47. ### Studying the convergence of a series with an arctangent of a partial sum

Greeting I'm trying to study the convergence of this serie I started studying the absolute convergence because an≈n^(2/3) we know that Sn will be divergente S=∝ so arcatn (Sn)≤π/2 and the denominator would be a positive number less than π/2, and because an≈n^(2/3) and we know 1/n^(2/3) >...
48. ### Convergence of a series involving ln() terms in the denominator of a fraction

good day I want to study the convergence of this serie and want to check my approch I want to procede by asymptotic comparison artgln n ≈pi/2 n+n ln^2 n ≈n ln^2 n and we know that 1/(n ln^2 n ) converge so the initial serie converge many thanks in advance!
49. ### Discussing the Convergence of a Series: Get My Opinion!

Good day I want to study the connvergence of this serie I already have the solution but I want to discuss my approach and get your opinion about it it s clear that n^2+5n+7>n^2+3n+1 so 0<(n^2+3n+1)/(n^2+5n+7)<1 so we can consider this as a geometric serie that converge? many thanks in advance
50. ### Problem with series convergence — Taylor expansion of exponential

Good day and here is the solution, I have questions about I don't understand why when in the taylor expansion of exponential when x goes to infinity x^7 is little o of x ? I could undesrtand if -1<x<1 but not if x tends to infinity? many thanks in advance!