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. J

    Finding the sum of a convergent Series

    Homework Statement Sum (n=1->infinity) of 2^(n+2)/3^n Homework Equations The Attempt at a Solution I have literally no idea how to attempt this. We beat into the ground the process of testing convergence and finding that, but not how to find the actual SUM. any help...
  2. G

    Air flow in Convergent divergent nozzle

    i understood the air flow properties variation in convergent divergent / condi nozzle or laval nozzle for subsonic flow based on the formula, for a incompressible flow : density * Area * velocity = constant as the total pressure in the flow is constant dynamic pressure is...
  3. H

    Are These Infinite Series Convergent or Divergent?

    Homework Statement I have been straining to find convergence or divergence of a few infinite series. I have tried everything I can think of; ratio test, root test, trying to find a good series for comparison, etc. Here are the formulas for the terms: #1 1 ------------- (ln(n))^ln(n)...
  4. F

    Uniformly convergent series and products of entire functions

    If the sum of a sequence of functions a_n converges uniformly, how is it that the product of 1+a_n converges uniformly? I know that this is true if the a_n are constants but how does this translate to functions?
  5. D

    Convergent Lenses: 2-Part Problem and Solution | Focal Point and Image Location

    Homework Statement This is a 2 part problem but I figure out the first part. Heres the 1st problem and the solution: 7- Given a convergent lens which has a focal point f. An object is placed at distance p = \frac{4}{3}f to the left of the lens. See the sketch. Solution: q1 = 4f, and is a...
  6. T

    Convergent Integral Solution | Determining Convergence with f(x) and g(x)

    Homework Statement Does \int_{0}^{\infty}\frac{dx}{1+\left(xsinx\right)^{2}} converge? I don't know if this is a legitimate solution. Any insight? Thanks Tal The Attempt at a Solution No. f(x)=\frac{1}{1+\left(xsinx\right)^{2}}\geq g(x)=\begin{cases}...
  7. L

    Series is no longer convergent

    Hi, "Given that sum x_n converges, where x_n are real, does sum (x_n)^3 necessarily converge?" My gut feeling is "no". When considering conditional convergent series. It may be that the cubing can increase the ratio of consecutive "groups of terms" (ie the terms in the series we consider as...
  8. U

    Real Analysis: product of convergent sequences

    Homework Statement suppose {an} and {bn} are sequences such that {an} converges to A where A does not equal zero and {(an)(bn)} converges. prove that {bn} converges. Homework Equations What i have so far: (Note:let E be epsilon) i know that if {an} converges to A and {bn}converges...
  9. S

    Determine whether the series is convergent or divergent.

    Hello, I have to determine whether the series converges or diverges. It is \Sigma (-1)^n * cos(Pi/n) where n=1 and goes to infinity. First I took the absolute value of the function and got the limit from n to infinity of cos(pi/n) and as a result I got 1 because cos(0)=1. However my...
  10. C

    Series - Convergent or Divergent?

    Is the series convergent or divergent? n=1 summation and it goes to infinity n!/2n!+1 [Infinite series] Homework Equations None. The Attempt at a Solution I have no idea.
  11. C

    Convergent Series Proof

    Homework Statement Let f:Z\rightarrowR be periodic such that f(x+a) = f(x) for some fixed a\geq1. Prove that \Sigma ^{infinity}_{x=1} \frac{f(x)}{x} converges if and only if \Sigma ^{a}_{x=1} f(x) = 0. Homework Equations n/a The Attempt at a Solution Ok, so I have a general...
  12. A

    (Dis)proof of a sequence of functions being uniformly convergent

    Homework Statement Let g_n : R -> R be given by gn (x) := cos2n (x), does gn' converge uniformly? Homework Equations The derivative is as follows; -2nsin(x)cos2n-1, which I have found converges pointwise to the 0 function. Formal definition of Uniform Convergence; For all e>0...
  13. P

    Converging to Zero: How to Solve Bijective Mapping Sequences Problem?

    Homework Statement Let f: N -> N be a bijective map. for n Є N a sub n = 1 / f(n) Show that the sequence (a sub n) converges to zero. Homework Equations The Attempt at a Solution Basically I have been stuck on this problem for hours now and have read through my notes and...
  14. P

    Need explanation on convergent

    http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations Could someone be nice to explain me the convergent condition as written in wiki ? I don't have an idea what it really means :frown: I am learning the basics only Thank you
  15. M

    Tychonoff's theorem and convergent subsequence

    Let X be the countably infinite product of closed unit intervals under the product topology. By Tychonoff's theorem, this space is compact. Consider the sequence \{ x_n \} , where x_k is the vector that is zero for all components except for the kth component, which is 1. Since this space...
  16. V

    Convergent series and their corresponding sequences (analysis course)

    If a_n >= 0 for all n, and the series a_n converges, then n(a_n - a_n-1) --> 0 as n --> infinity. Prove or disprove the statement using a counterexample. I know that the statement is false...I am just having terrible difficultly finding a counterexample...
  17. M

    Bounded sequence that diverges, convergent subsequence

    Homework Statement Let (sn) be a sequence in R that is bounded but diverges. Show that (sn) has (at least) two convergent subsequences, the limits of which are different. Homework Equations The Attempt at a Solution I know that a convergent subsequence exists by...
  18. K

    Bounded sequence, convergent subsequence

    Homework Statement Asssume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a. Show that (an) must converge to a. Homework Equations The Attempt at a Solution If the subsequence converges to a we have , we have...
  19. C

    Explaining Convergent Sequences w/ Examples

    Could anyone please explain "convergent sequence" with example.
  20. S

    Showing a (complex) series is (conditionally) convergent.

    I've been reading a complex analysis book which had an example showing \sum^\infty_{n=1}1/n \cdot z^n is convergent in the open unit ball. I'm now looking at the case when |z| = 1. Clearly z = 1 is the divergent harmonic series, but i know this series is in fact convergent for all other |z| =...
  21. B

    Is function of convergent sequence rate of change equal to derivative?

    Given a convergent sequence x_n \rightarrow x and a function f, is lim_{n \rightarrow \infty} \frac{f(x_n) - f(x_{n-1})}{x_n - x_{n-1}} = f'(x) ? I believe it it is, but I haven't been able to figure out how to prove it. Does anyone know of a proof or counter-example? And probably should...
  22. C

    Convergent Subsequences in Compact Metric Space

    Homework Statement Suppose that (x_n) is a sequence in a compact metric space with the property that every convergent subsequence has the same limit x. Prove that x_n \to x as n\to \infty Homework Equations Not sure, most of the relevant issues pertain to the definitions of the space...
  23. R

    Proving Convergence of Averages of a Convergent Sequence

    Homework Statement Show that if (x_n) is a convergent sequence, then the sequence given by the averages y_n = (x_1+ x_2 +...+ x_n)/n also converges to the same limit. Homework Equations The Attempt at a Solution I think I need to show that for n >= N, |y_n - lim(x_n)| <...
  24. F

    Convergent Series: sin(kx)/ln(k)

    Homework Statement Is this series convergent for all real x: \sum^{\infty}_{k=2}\frac{sin(kx)}{ln(k)} Homework Equations The Attempt at a Solution This series is less than \frac{1}{ln(2)}\sum^{\infty}_{k=2}sin(kx) which is less than \frac{\pi}{x ln2}. So, the series is bounded for all x...
  25. N

    Is sexual reproduction an example of convergent evolution?

    Second thread on the evolution of like, in the Archaen and Proterozoic eons. There are many kinds of eyes, and many have arisen quite independently; the classic example of convergent evolution. Did sexual reproduction arise more than once, independently? Is it too an example of convergent...
  26. T

    Product of convergent infinite series converges?

    Homework Statement Given two convergent infinite series such that \sum a_n -> L and \sum b_n -> M, determine if the product a_n*b_n converges to L*M. Homework Equations The Attempt at a Solution If know that if a_n -> L this means that the sequence of partial sums of a_n = s_n...
  27. R

    Convergent Series: Find Sequence of Positive Reals

    Homework Statement Find a sequence (an) of positive real numbers such the sum of an from 1 to infinity is convergent but the number of k such that a(k+1)>ak divided by n tends to 1 as n tends to infinity. Homework Equations The Attempt at a Solution I don't have a clue how to find...
  28. V

    Find the sum of the convergent series 4-2+1-1/2

    find the sum of the convergent series 4-2+1-1/2 ... Homework Statement find the sum of the convergent series . 4-2+1-1/2... Homework Equations sum of r^n |r|< 1 --> 1/(1-r)= sum The Attempt at a Solution 4(1-1/2+1/4-1/8...) I can see a pattern here where the denominator is going 2,4,8 so...
  29. M

    Convergent Series: ln(n/(n+1))

    Homework Statement ∑ ln((n)/(n+1)) I was assuming this would be \infty/\infty and if I divide through by n it gives me 1/1 or 1 so would this just be divergent? Homework Equations The Attempt at a Solution
  30. K

    Finding if a series is convergent.

    Finding if a series is convergent-Answered Homework Statement Find for which values of K is the fallowing series convergent. \sum((n!)2)/((kn)!) where: N is the variable. K is a constant or a list of constant (eg. "(2,91]") Homework Equations I believe the ratio test, which...
  31. Z

    Use comparison theorem to show if integral is convergent or divergent

    Homework Statement int (e^-x)/(x)dx from 0 to infinity Determine if integral is convergent or divergent2. The attempt at a solution I assume because the bottom limit is 0 and there is an x in the bottom of the integral that this is going to be divergent but I still have to use the...
  32. B

    Pair of convergent subsequences

    Let (xn) be a bounded sequence that diverges. Show that there is a pair of convergent subsequences (xnk) and (xmk), so that lim_{k\rightarrow\infty} \left|x_{nk} - x_{mk}\right| > 0
  33. B

    Analyzing Convergent Series w/Supremum & Integral Norms

    I have been thinking about this problem: Determine whether the following series are convergent in \left(C[0,1],||\cdot ||_{\infty}\right) and \left(C[0,1],||\cdot ||_{1}\right). when f_n(t)=\frac{t^n}{n} In the supremum norm, this seems pretty straightforward, but in the integral norm I am...
  34. Z

    Probability: Infinite Convergent Series and Random Variables

    I have a random variable problem. I need to prove that my equation I came up with is a valid probability mass function. In the problem, I came up with this for my probability mass function: \Sigma 12/(k+4)(k+3)(k+2) Maple says that this does in fact converge to 1, so it's valid...
  35. L

    Convergent or Divergent Integral: Comparison Theorem

    Homework Statement Determine if the following is improper and convergent, improper and divergent, or proper \int \frac{dx}{\sqrt[3]{x^2 - 7}} from 8 to infinityThe Attempt at a Solution Since I don't know how to integrate...
  36. Somefantastik

    Proving Convergence of Series in a Given Norm

    Given a sequence, how does one prove that the associated series in convergent or not, in a given norm? For example, \sum_{k=0}^{\infty}a_{k} in ||\cdot|| The process to do this is not in my book; I'm told how to determine whether a series is cauchy, but I'm not sure how to use that to...
  37. S

    Convergence of a Monotonic Sequence: Proof and Example

    Homework Statement If a_{1} = 1 and a_{n+1} = (1-(1/2^{n})) a_{n}, prove that a_{n} converges. Homework Equations NONE The Attempt at a Solution I am confident about my attempt, I just want it checked. Thanks. First show that a_{n} is monotone: a_{n} = {1, 1/4, 21/32...
  38. K

    Cauchy sequence with a convergent subsequence

    Homework Statement Theorem: In a metric space X, if (xn) is a Cauchy sequence with a subsequence (xn_k) such that xn_k -> a, then xn->a. Homework Equations N/A The Attempt at a Solution 1) According to this theorem, if we can show that ONE subsequence of xn converges to a, is that...
  39. A

    Convergent and Divergent Integrals

    I had a question regarding convergent and divergent integrals. I want to know the "exact" definition of an improper integral that converges. Wikipedia states that For a while, I took that as a valid answer and claimed that any integral that has a finite answer must be convergent. However, I...
  40. R

    Determine whethere the following series is convergent or divergent

    Homework Statement Determine whether the following series is convergent or divergent: \frac{1}{2^2}+\frac{2^2}{3^3}+\frac{3^3}{4^4}+... I rewrite it as: \sum_{n=1}^{\infty} \frac{n^n}{(n+1)^{n+1}} Homework Equations The Attempt at a Solution I stopped. I can not do anything.
  41. A

    Probability of Interbreeding via Convergent Evolution

    I realize this might not be likely, but I'm sure it could happen. I'm just trying to figure out how improbable this scenario may be: How likely is it that two different species could undergo convergent evolution to such an extent that members of the different species could interbreed? In...
  42. L

    Is a series is convergent or divergent

    Homework Statement Determine the convergence or divergence of the series. If the series is convergent, find its sum. Justify each answer. (n=1, to infinity) \sum(7/9 + n^5) Help please? I missed a lot of school recently from being sick and need help with this!
  43. A

    Showing that the set of convergent sequences is not separable

    Hi, I'm looking to show that the metric space of convergent complex sequences under the sup norm is not separable; that is what I assume it is since I cannot find a way to prove that it is separable (I am unable to find any dense subsets). A set of complex sequences convergent to a certain...
  44. G

    I know it is convergent by I cannot determine why

    Homework Statement 0 to infinity sum of 6/(4n-1)-6/(4n+3) Determine if the series is convergent or divergent. The Attempt at a Solution I know it is convergent by I cannot determine why.
  45. A

    Searching for a particular kind of convergent sequence

    I want an example of a complex sequence (x_n) which converges to 0 but is not in ℓ^p, for p\ge 1 i.e. the series \sum |x_n|^p is never convergent for any p\ge 1. Can someone provide an example please?
  46. A

    Convergent Sequence: t^n/(n Factorial) Limit as n->∞?

    i have a sequence an= t^n / (n factorial). I know that the infinite series of it converges to zero, but i need to know if the limit of an goes to zero or not , as n goes to infinity. Thanks
  47. A

    Serie convergent to cosec

    Hi all! I found on a book of QFT in curved spacetime (Birrel and Davies, pag 53) the following identity cosec^2 \pi x = \frac{1}{sin^2 \pi x} = \pi^{-2} \sum_{k=-\infty}^{+\infty} \frac{1}{(x-k)^2} Can anyone help to derive it or give some reference to a book for the proof. I have no idea...
  48. G

    Proving Convergence of a Sequence with a Given Upper Bound

    Homework Statement Proove rigorously that if (a_{n} is a real convergent sequence with lim_{n\rightarrow \infty} a_{n} = a and for each n=\in N, a_{n} < 6, then a \leq 6 Homework Statement Homework Equations The Attempt at a Solution Let \epsilon > 0 we need to find n_{0} \in N such that...
  49. E

    Convergent -divergent series

    Suppose the series \sum a_{n} diverges to +\infty, Then if the series does not diverge to infinity it means that the series converges, and consequently the statement : if \sum a_{n} diverges ,then \sum b_{n} diverges,is equivalent to : if \sum b_{n} converges ,then \sum a_{n} converges??
  50. T

    Convergent Series Homework: For What Values of p Does It Converge?

    Homework Statement For what values of p does the series [1/1^p - 1/2^p + 1/3^p - 1/4^p +... converge? Homework Equations The Attempt at a Solution I believe that this series converges for all p \in N because the sequence of a_n's is nonincreasing and converges to 0. I am not...
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