What is Convergent: Definition and 334 Discussions

Convergent evolution is the independent evolution of similar features in species of different periods or epochs in time. Convergent evolution creates analogous structures that have similar form or function but were not present in the last common ancestor of those groups. The cladistic term for the same phenomenon is homoplasy. The recurrent evolution of flight is a classic example, as flying insects, birds, pterosaurs, and bats have independently evolved the useful capacity of flight. Functionally similar features that have arisen through convergent evolution are analogous, whereas homologous structures or traits have a common origin but can have dissimilar functions. Bird, bat, and pterosaur wings are analogous structures, but their forelimbs are homologous, sharing an ancestral state despite serving different functions.
The opposite of convergence is divergent evolution, where related species evolve different traits. Convergent evolution is similar to parallel evolution, which occurs when two independent species evolve in the same direction and thus independently acquire similar characteristics; for instance, gliding frogs have evolved in parallel from multiple types of tree frog.
Many instances of convergent evolution are known in plants, including the repeated development of C4 photosynthesis, seed dispersal by fleshy fruits adapted to be eaten by animals, and carnivory.

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  1. C

    Sequence is norm convergent implies it's strongly convergent

    If a sequence of operators \{T_n\} converges in the norm operator topology then: $$\forall \epsilon>0$$ $$\exists N_1 : \forall n>N_1$$ $$\implies \parallel T - T_n \parallel \le \epsilon$$ If the sequence converges in the strong operator topology then: $$\forall \psi \in H$$...
  2. S

    Is this sequence divergent or convergent?

    Homework Statement I'm trying to find out whether or not this sequence diverges or converges. If it converges, then what's the limit. {4+sin(1/2*pi*n)} The Attempt at a Solution This one is a bit confusing to me since sin oscillates between 1 and -1. So if you plug in (pi*infinity)/2, that...
  3. S

    Proving Convergent Sequence of x^t to 0

    Homework Statement Show, from the definition of what it means for a function to converge to a limit, that the sequence ##\left\{x^t\right\}_{t=1}^{\infty}## with ##x^t = \frac{2t+5}{t^2+7}## converges to ##0## as ##t## goes to infinity. Homework Equations A sequence converges to ##x^0 \in X##...
  4. M

    Convergent or Divergent: Is this a Convergent Series?

    Mod note: Moved from a homework section. 1. Homework Statement this is my lecturer's notes, he says it is a divergent series, but this seems like an obvious convergent series to me.. could someone verify? Homework Equations https://www.dropbox.com/s/mc5rth0cgm94reg/incorrect maths.png?dl=0...
  5. Cake

    Show that the series is absolutely convergent

    Homework Statement Show that ##\sum \frac {cos(\frac{n\pi} {3})} {n^2}## is absolutely convergent, and therefore convergent Homework Equations Comparison test to 1/n^2 The Attempt at a Solution So to be absulutely convergent the absolute value of the series needs to be convergent. So we...
  6. StrangeCharm

    Finding the Limit of a Convergent Sequence

    Homework Statement Determine whether the sequence converges or diverges. If it converges, find the limit. Here's the sequence: http://www4a.wolframalpha.com/Calculate/MSP/MSP89541ea2ag9dg617bcd6000050d52e94i67ei593?MSPStoreType=image/gif&s=39&w=66.&h=44. Homework Equations N/A The Attempt at...
  7. D

    Unique Property of Convergent Series?

    I was goofing around with Mathematica and found that Sum_(k>=1)(sin(k)/k)=Sum_(k>=1)(sin(k)/k)^2. In other words a convergent series such that if you square each of its terms the sum is the same. Question is: is this a unique property or are there other convergent series with the property? Cheers.
  8. J

    Fixed point iteration, locally convergent

    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...
  9. Newb_Aero_Ninja

    Convergent Subsonic Ramjet Utilizing Shockwave Compression

    Hi, I am working on investigating an idea I proposed regarding a ramjet that operates in subsonic flow (of a fixed speed) with a convergent intake. That utilizes the pressure immediately behind a standing shock-wave for compression. I have posted a link to my initial report here and I now need...
  10. C

    Why is this integral not convergent? Could be a technicality

    Homework Statement Wolfram alpha and integral calculator say zero, but my school's HW on webworks does say it's divergent. I don't suppose it's wrong either, since the issue is probably a technicality Arguably, on desmos.com the graph is pretty damn near odd. Homework Equations Int(-inf)(inf)...
  11. J

    Calculating Mass Flow Rates and Mach Numbers in a Convergent-Divergent Nozzle

    Hi, I hope that, given I've sourced what I can, you may be able to help? I'm Currently working on a lab report for my Aeroengines unit based on Convergent-Divergent nozzle (http://imgur.com/HQML2Au). From the drawing, I have been provided D1, D2, D3, D4, T1, T2, T3, T4, P1, P2, P3, Velocity...
  12. C

    Every sequence has a convergent subsequence?

    I'm not sure if this is true or not. but from what I can gather, If the set of Natural numbers (divergent sequence) {1, 2, 3, 4, 5,...} is broken up to say {1}, is this a subsequence that converges and therefore this statement is true?
  13. A

    MHB Can a Numerical Series Converge to a Functional Series?

    hi everyone ! i have a question , Can a numerical convergent series be a convergent functional series? I know only the other way that a convergent functional series can be a convergent numerical series because i take a function as a constant so i have a numerical series and it converges.
  14. G

    Sequence (n)/(n^n) Convergent or Divergent and Limit?

    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...
  15. Fredrik

    Cauchy sequences and absolutely convergent series

    Homework Statement I want to prove that if X is a normed space, the following statements are equivalent. (a) Every Cauchy sequence in X is convergent. (b) Every absolutely convergent series in X is convergent. I'm having difficulties with the implication (b) ⇒ (a). Homework Equations Only...
  16. H

    Solve Convergent Series with Mathematica: Pi Squared/8

    Hi, Mathematica is telling me the value of this series, but I can't figure out how to do it on paper. Can someone please explain? $$\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}$$
  17. S

    Prove that a convergent sequence is bounded

    Homework Statement The problem and solution are attached as TheProblemAndSolution.jpg. Homework Equations Definition of the limit of a sequence. The Attempt at a Solution I understand how P = ϵ + |A| can be seen as an upper bound that proves that the sequence is bounded, but for the last bit...
  18. U

    Determine if the series is convergent or divergent and radio test

    Homework Statement 1+\dfrac{2^2}{2!}+\dfrac{3^2}{3!}... \infty The Attempt at a Solution t_n = \dfrac{n^2}{n!} \\ \dfrac{n}{(n-1)(n-2)...1} I tried applying the Ratio Test but couldn't find another function which would give me a finite limit when divided by that function.
  19. I

    MHB How to determine whether an integral is convergent or divergent

    Determine whether the integral is convergent or divergent. Evaluate those that are convergent. $\int_{0}^{9} \ \frac{1}{\sqrt[3]{x-1}},dx$ $\int_{-\infty}^{\infty} \ \cos\left({\pi t}\right),dt$ how do i determine whether it's conver/diver?
  20. B

    Series Convergence Test for ∑ 5^n/(4^n +3)

    Homework Statement Which of the series, diverge or converge ∑ 5^n/(4^n +3 ) Homework Equations The Attempt at a Solution Taking the limit as n→∞ we have (5^n ln 5)/ (4^n ln 4) , my question is here how does it become like this, which part am I missing here?
  21. kelvin490

    Is the Sequence xn=1/(n-2) Convergent for All Values of n?

    Suppose there is a sequence xn=1/(n-2). We know we n tend to infinity the sequence tends to zero. But at n=2 it is equal to infinity. Is this sequence convergent? There is also a theorem that all convergent sequence are bounded for every n. But the sequence above is not bounded at n=2...
  22. S

    MHB Determine if the SERIES DIVERGES, absolutely convergent, conditionally convergent(IV)

    determine if series is absolutely convergent, conditionally convergent, or divergent \sum^{\infty}_{n = 1} (n^2 + 9)(-2)^{1-n} which i turned into \sum^{\infty}_{n = 1} (n^2 + 9)(-2)^{-n+1} so using the ratio test I got: \frac{((n+1)^2 + 9)(-2)^{-n})}{n^2 + 9 * (-2)^{1-n}} which ended...
  23. S

    MHB Question about determining if the series is convergent or divergent

    Determine if the positive term series is convergent or divergent \sum^{\infty}_{n = 1} \frac{n + cosn}{n^3 + 1} can't I just ignore the cosn and look at it like this: \sum^{\infty}_{n = 1} (-1)^n \frac{n}{n^3 + 1} Then can't I just look at it as n--> \infty and see that I end up with...
  24. G

    Series: Convergent or Divergent?

    Homework Statement Is the following convergent or divergent? ln(2(n+1))-ln(2n) Homework Equations comparison test The Attempt at a Solution I put it into the form ln(1+(1/n)), but I don't understand what to use as Bn for the comparison test. (which is what wolfram alpha uses)
  25. T

    Explicit formula for a convergent series

    Hello PF. Homework Statement A function f is defined by f(x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6... Find the radius of convergence of the series and the explicit formula for f(x).Homework Equations The Attempt at a Solution I know that the formula for the series is going to be similar to the...
  26. S

    Help with determining if a series is convergent or divergent question

    Homework Statement Problem is attached in this post. Homework Equations Problem is attached in this post. The Attempt at a Solution I know how to find the sum of this series, but I don't know what method to use in order to prove that this series converges. Also I don't understand how I'm...
  27. C

    Limits of convergent sequences

    Homework Statement an= (n/n+2)^n ANS: 1/e^2 The Attempt at a Solution I was told this was convergent and I need to find the limit of the sequence. How do I do this, as I seem to keep getting that this is divergent. Isn't it divergent to infinity? Or am I missing something?
  28. C

    Limits of convergent sequences

    Homework Statement an= (n/n+2)^n The Attempt at a Solution I was told this was convergent and I need to find the limit of the sequence. How do I do this, as I seem to keep getting that this is divergent. Isn't it divergent to infinity? Or am I missing something?
  29. U

    Series: Divergent or Convergent ?

    How can I tell if the following series is Divergent or Convergent: ∑( e^(6pi*n) sin^2(4pi*n) ) the sum limits are from -infinity to infinity
  30. C

    Divergent or Convergent? Evaluating an Integral with Exponential Function

    Homework Statement Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. ∫ from negative infinity to infinity of (x^8*e^-x^9) The Attempt at a Solution The answer is diverged to infinity. But I got that by guessing. Can someone explain to me why...
  31. A

    Find all possible real numbers such that the series is convergent.

    Homework Statement Is there a real number c such that the series: ∑ (e - (1+ 1/n)^n + c/n), where the series goes from n=1 to n=∞, is convergent? The Attempt at a Solution I used the ratio test by separating each term of the function as usual to find a radius of convergence, but that doesn't...
  32. 9

    Convergent limits for sequences: picture terms

    A limit of a sequence is definitely convergent if: If for any value of K there is an N sufficiently large that an > K for n > N, OR for any value of K there is an N sufficiently large that an<±K for n > N My only question is what exactly are K, N, an and n? What values are they? How would...
  33. 9

    Sequence of 3/4^(2k). Show is convergent, find sum

    Homework Statement Consider the sequence ak = (3)/(4^(2k)). Show is convergent, find sum. Please check work. Homework Equations ak = 3/(4^2k) let s {n} be the series associated with the sequence. Cannot write summation notation here, but k starts at 1 (k = 1 on bottom) and infinity...
  34. 9

    Convergent and Divergent Sequences

    Homework Statement Please look over my work and tell me if I did something wrong. Suppose Bn is a divergent sequence with the limit +∞, and c is a constant. Prove: lim cBn -> ∞ = +∞ for c > 0 Homework Equations N/A The Attempt at a Solution lim Bn -> ∞ = means that for some value K >...
  35. J

    CONVERGENT nozzles contraction ratio and length

    I am currently designing a convergent nozzle for use in experiments and wanted to check something: Will the pressure and density of the flow always expand to ambient (following the isentropic relations) when it reaches the nozzle exit regardless of nozzle contraction ratio and length? Any...
  36. F

    Series Convergence Tests: Arctan and Grouped Terms

    Homework Statement 1. Determine if arctan(7+1/n)-arctan(7) converges or diverges 2. Determine if 2/1-1/2-1/3+2/4-1/5/-1/6+2/7... converge or diverge Homework Equations series tests The Attempt at a Solution 1.My gut instinct is to do limit comparison test w/ 1/n, and it...
  37. G

    Divergent limit + divergent limit = convergent limitIs it possible?

    Homework Statement As a part of Method of Frobenius, I am encountered with the following problems: Evaluate the following limits: Q1. \stackrel{limit}{_{x→0}}\frac{1-2x}{x} Q2. \stackrel{limit}{_{x→0}}\frac{x-1}{x} Q3. \stackrel{limit}{_{x→0}}\frac{1-2x}{x}+\frac{x-1}{x} In context of the...
  38. N

    Is this series convergent or divergent.

    Homework Statement Ʃ ne(-n2) Homework Equations The Attempt at a Solution I used the ratio test and wanted to know if the way I did it is correct or not |a(n+1) / a(n)| n+1 (e(-n2 -2n-1)) / n e(-n2) Now e-n^2 cancels and we get limn→∞ n+1/n * 1/(e2n)(e) After you take the limits you get...
  39. M

    Prove Convergence of $\sum_{n=1}^{\infty} a_n$ and $\sum_{n=1}^{\infty} b_n$

    Homework Statement . Let ##\{a_n\}_{n \in \mathbb N}## a sequence of real numbers such that ##lim_{n \to \infty} a_n=0## and let ##b_n=a_n+2a_{n+1}-a_{n+2}##. Prove that ##\sum_{n=1}^{\infty} a_n## is convergent iff ##\sum_{n=1}^{\infty} b_n## is convergent. The attempt at a solution...
  40. F

    Ord function and convergent in Qp

    Homework Statement Solve the following : a) Show that ordp((p^n)!)=1+p+p^2+p^3+...+p^(n-1) b)For which values of p does the following series converge in Qp? 1)1+(15/7)+(15/7)^2+(15/7)^3+... 2)1!+2!+3!+4!+... 2. The attempt at a solution For a) I want to to count how...
  41. C

    Prove that all convergent sequences are bounded

    was looking at a proof of this here: http://gyazo.com/8e35dc1a651cec5948db1ab14df491f8 I have two questions, why do you set K = max of all the terms of the sequence plus the 1 + |A| term? Why do you need the absolute value of all the terms? i.e. why |a_1| instead of |a_1|?
  42. P

    Finding a convergent subsequence does the sequence need to be bounded

    Homework Statement 2.11. Determine (explicitly) a convergent subsequence of the sequence in R2 given for n = 1; 2; : : : by xn =(e^{n}sin(n\pi/7),((4n+3/3n+4)cos(n\pi/3)) I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I...
  43. Fernando Revilla

    MHB Can a sequence without a convergent subsequence have a limit of infinity?

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  44. M

    Convergent Series: Am I and The Book Wrong or is Wolfram?

    In my book it says that the series (1+x)^n converges for x<1. However I put n = -1 and wolfram says that the series does not converge. However if I let x = 1/y where y>1 then the expansion of (1+1/y)^-1 is equal to: (which I will define as (SERIES 1)) 1 - y + (1/y)2 - (1/y)3 + (1/y)4...
  45. STEMucator

    Prove every convergent sequence of real numbers is bounded &

    Homework Statement The question : http://gyazo.com/7eb4b86c61150e4af092b9f8afeaf169 Homework Equations Sup/Inf axioms Methods of constructing sequences ##ε-N## ##lim(a_n) ≤ sup_n a_n## from question 5 right before it. I'll split the question into two parts. The Attempt at a...
  46. D

    Convergent and divergent series

    Is it posible that the product of two different divergent series be a convergent serie?
  47. M

    Series Test for convergent and divergent

    Homework Statement Ʃ √n/(ln(n))^2 from n=2 to ∞ Homework Equations Series Test for convergent and divergent The Attempt at a Solution I tried doing ratio test and gotten [√(n+1)*(ln(n))^n] / [(ln(n+1))^(n+1) * √n] to find the limit, do...
  48. L

    Find sum of convergent series: 2/[(4n-3)(4n+1)]

    1. Find the sum of the convergent series: ∞ Ʃ 2/[(4n-3)(4n+1)] n=12. Hm... Okay, so I started with the nth term test, and the denominator gets huge very fast. So I'm pretty sure it goes to zero. So that tells us nothing other than that it does not FOR SURE diverge. Since it has no n in the...
  49. S

    Physics of a convergent nozzle

    Hello, Imagine a convergent nozzle; static pressure at exit is atmospheric. The fluid is air. The pressure on the pressurized side is P. goal #1: achieve nozzle exit velocity somewhat below sonic goal #2: have P as high as possible. Is this possible to achieve through the geometry of...
  50. M

    Series Test for convergent and divergent

    Hello. I.m struggling to understand how to test for convergent and divergent. ∞ Ʃ (n/(n+1))^(n^2) n=1
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