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. R

    Convergent or divergent, p-series.

    Homework Statement Problem is to determine if this is convergent or divergent: n = 1 E infinity (27 + pi) / sqrt(n) Homework Equations p-series test? The Attempt at a Solution I was looking at this problem, It looks as if the p-series may apply, it is continuous, decreasing...
  2. S

    Determine if a sequence {an} is monotonic, bounded, convergent

    Determine whether the sequence {an} defined below is (a) monotonic (b) bounded (c) convergent and if so determine the limit. (1) {an}=(sqrt(n))/1000 a) it is monotonic as the sequence increase as n increases. b) it's not bounded (but I'm not sure why) c) divergent since limit doesn't...
  3. M

    Convergent series - What is this N thing?

    I am currently reading through my notes and found the convergent series to be defined. lxn - Ll is less than (epsilon) whenever n is greater or equal to N ...i have looked on wikpedia and a few other web sites and i am not making any sense of what this N is... wikipedia says - 'a series...
  4. D

    Discover Radius of Convergent for the Series: (1-2^n)(ln(n))x^n - Solved!

    [Solved] Radius of Convergent Homework Statement Find the radius of convergent for \sum_{n=1}^\infty (1-2^n)(ln(n))x^n Homework Equations \frac {1}{R} = L = \lim \frac{a_{n+1}}{a_n} The Attempt at a Solution lim \frac {(1-2^{n+1})(ln(n+1)}{(1-2^n)(ln(n))} = L lim...
  5. D

    Is \sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n+1}-\sqrt{n}}{n} absolutely convergent?

    Homework Statement Decide if \sum_{n=1}^{\infty}(-1)^n\frac{\sqrt{n+1}-\sqrt{n}}{n} is convergent and if it is, is it absolutely convergent or conditionally convergent? The Attempt at a Solution I'm pretty sure that the \lim_{n\rightarrow\infty} a_n = 0 Am I supposed to use...
  6. M

    Counterintuitive Convergent Series

    [SOLVED] Counterintuitive Convergent Series Homework Statement One of my new textbooks in mathematical analysis makes a very strange claim (not sure if it was a true claim or some random historical anecdote) for a convergent series in one of its short sections on the history of mathematics...
  7. K

    Prove that sin (n^2) + sin (n^3) is not a convergent

    Prove that \sin (n^2) + \sin (n^3) is not a convergent sequence.
  8. A

    Determine whether the series is convergent

    Homework Statement Determine whether the series \sum_{n=2}^{\infty}a_n is absolutely,conditionally convergent or divergent a_n=\frac{(-1)^n}{\sqrt{n}(\frac{2n}{n+1})^\pi} The Attempt at a Solution from Abel's test.c_n=\frac{(-1)^n}{\sqrt{n}}is convergent.and...
  9. C

    Absolute Convergent Series proof

    Homework Statement Let Sumation "a sub n" be an absolutely convergent series, and "b sub n" a bounded sequence. Prove that sumation "a sub n"*"b sub n" is convergent. (sorry fist time on this site and can't use the notation.) Homework Equations Theorem A: that states sumation "a sub n"...
  10. S

    Bounded sequence implies convergent subsequence

    How can you deduce that nad bounded sequence in R has a convergent subsequence?
  11. L

    Convergence of Sequence x_n = (1/2)(x_n + 2/x_n)

    Hello, I have a question i can figure out. THE QUESTION: Show that \sqrt{2}, \sqrt{2\sqrt{2}}, \sqrt{2\sqrt{2\sqrt{2}}}, ... converges and find the limit.From what I see, the first term is root 2, the second term will be the root of 2 times the first term making it larger than the first...
  12. J

    This sequence has no convergent subsequence?

    Homework Statement Let {x_n} be a sequence in a metric space such that the distance between x_i and x_{i+1} is epsilon for some fixed epsilon > 0 and for all i. Can it be shown that this sequence has no convergent subsequence? Homework Equations None. The Attempt at a Solution...
  13. P

    Convergent sequence of functions?

    Homework Statement How do you show a sequence of functions in terms of n is convergent (in general)? The Attempt at a Solution Do you presuppose the value of the function say v then show d(fn,v)->0 for n large? v is determined by looking at the sequence of functions and guessing...
  14. T

    Is this product convergent

    Following Euler if we define the product: (x-2^{-s})(x-3^{-s}) (x-5^{-s})(x-7^{-s})...=f(x) taken over all primes and s > 1 ,what would be the value of f(x) ?? i believe that f(x,s)=1/Li_{s} (x) (inverse of Polylogarithm) however I'm not 100 % sure, although for x=1 you get the inverse...
  15. Loren Booda

    Most rapidly convergent reciprocal prime series equal to 1

    Consider the series with ascending (but not necessarily sequential) primes pn, 1/p1+1/p2+1/p3+ . . . +1/pN=1, as N approaches infinity. Determine the pn that most rapidly converge (minimize the terms in) this series. That set of primes I call the "Booda set."
  16. A

    Does Non-Absolutely Convergent Series Imply Convergence of Product Series?

    The problem states: Suppose \sum a_n and \sum b_n are non-absolutely convergent. Show that it does not follow that the series \sum a_n b_n is convergent. I tried supposing that the series \sum a_n b_n does converge, to find some contradiction. So the series satisfies the cauchy criterion and...
  17. S

    Solve Convergent Series Problem - Help Appreciated

    Attahced is a file of a problem I am trying to solve. Thanks for any help
  18. K

    Proving/Disproving Sequence Claims: a Convergent Sequence Example

    This is a question on a recent assignment that I can't figure out. I think if I understood the first part, I could get the rest. Let {a_n} be a convergent sequence with limit L. Prove or provide counter examples for each of the following situations. Suppose that there exists a number N...
  19. S

    Convergent matrice=divergent constant ?

    I am doing research into brain systems. Does anyone have any examples or links to existing equations where a a convergent matrice is the reverse of a divergent constant, such as log or phi ?
  20. A

    Convergence of Integral with Divergent Function at 0+

    I want to know if the integral \int_0^{\infty} dx/(4x^3 + x^(1/3)) is convergent or divergent?Thanks
  21. electronic engineer

    Bounded sequence as convergent

    Some rule says that not all bounded sequence must be convergent sequence , one example is the sequence with general bound: Xn=(-1)^n could anyone help?! thanks in advance!
  22. B

    Do These Mathematical Series Converge?

    (This isn't homework :redface:) Does this series converge? {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {\frac{1}{k}} } \right)^{ - 1} } \right]} } Does this series converge? {\sum\limits_{n = 1}^\infty {\left[ {\left( {\sum\limits_{k = 1}^n {k!} } \right)^{ - 1} }...
  23. T

    Is sin convergent or divergent

    I just have a quick question, is cos and sin divergent or convergent? I keep getting mixed results from different sources. I know that both functions oscillate so on the interval [0, infinity) they both diverge. But for some of my homework problems relating to improper integrals, the book...
  24. S

    Negate "f has limit L at c": f does not have limit L at c

    I found this in another threat however i do not know wat he means by convergent sequences. Is something like when u trying to take the limit at an ASYMPTOTE of a fuction? i know that the limit doesn't not exist( or goes to infinitive i cannot recall) is that wat he means by convergent sequence...
  25. W

    Need to find a convergent value

    Given this sum s = \sum_{k = 1}^{{\frac{x}{j}} - 1} k^{n}j^{n+1} x and n are constants and x/j is a positive integrer and k is an integrer To what value s converges as {j}{\rightarrow}{0} ? Edit: I have found that the awnser is \frac{x^{n+1}}{n+1}, but i do not know how to obtain this...
  26. happyg1

    Proving "No Convergent Subsequence -> |s| Diverges to Infinity"

    Hi, Here is the question: Prove that if the sequence {s} has no convergent subsequence then {|s|} diverges to infinity. To me, this seems so easy, but I'm having a really hard time putting it down in a rigorous manner. My thoughts are: every convergent sequence has a convergent...
  27. S

    What is the ratio test for proving absolute convergence of a series?

    hello all well i think I am kind of brain dead, iv been workin on a lot of problems over the last few days, I can't see anything obvious anymore, well this shall be the last one for today (i hope), anyway here it is, suppose that for some x\not= 0 , the series \sum_{n=1}^{\infty} a_n...
  28. S

    Solving Convergent Series Problem: Proving Limit of nan is 0

    hello all iv been workin on this problem its kind of awkward check it out {an} is a decreasing sequance, an>=0 and there is a convergent series Sn with terms an we need to prove that the limit of nan is 0 i first started of a sequence bn=an+1+an+2+...+a2n then I showed that the limit...
  29. Loren Booda

    Convergent cosmological evolution

    There are many examples of convergent evolution here on Earth, such as the separate development of wings on birds, bats and insects. I propose that the midpoint of evolution be marked by an equivalence between convergent and divergent adaptations. For the universe as a...
  30. cepheid

    Calculating Convergent Series: Tips for $\sum_{n=1}^{\infty} n^2 w^n$

    Does anyone have tips on how to sum the following series? \sum_{n=1}^{\infty} n^2 w^n Region of convergence is for |w| < 1
  31. N

    Convergence of Harmonic Series with Omitted 9s in Denominator

    In the harmonic series 1+1/2+1/3+1/4+... we omit expressions which contain digit 9 in denominator (so we omit e.g. 1/9, 1/19, 1/94, 1/893, 1/6743090 etc.). Proof that such series is convergent. Have You got any idea how to solve this problem? Thanks a lot for help
  32. P

    What is the Correct Integral Value for Divergent or Convergent?

    \int_9^{inf} \frac{1}{x^{6/5}} first thing i did was found the integral of the function \frac{5}{x^{-1/5}} then plug in inf(i will name it b) and 9 \frac{5}{b^{-1/5}} - \frac{5}{9^{-1/5}} now i will find the lim -> inf well for \frac{5}{9^{-1/5}}, it's equal to 7.759 now for...
  33. D

    Let's suppose that both sequences are convergent

    Consider the following statement: If \left\{ a_n \right\} and \left\{ b_n \right\} are divergent, then \left\{ a_n b_n \right\} is divergent. I need to decide whether it is true or false, and explain why. The real problem is that I checked the answer in my book; it's false, but I...
  34. S

    Convergent field, divergent potential?

    If you want to calculate the electric field at a distance r from a line of infinite length and uniform charge density you could one of three things: 1. Employ symmetry and Gauss' law. 2. Use superposition and integrate from minus to plus infinity along the rod. 3. Integrate to find the...
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