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. G

    Convergent sequences in the cofinite topology

    How can you identify the class of all sequences that converge in the cofinite topology and to what they converge to? I get the idea that any sequence that doesn't oscillate between two numbers can converge to something in the cofinite topology. Considering a constant sequence converges to the...
  2. S

    Using comparison theorem to show if an integral is convergent or divergent

    Homework Statement use the comparison theorem to show that the integral of e^(-x^2) from 0 to infinity is convergent.Homework Equations None The Attempt at a Solution In class we have never dealt with using the comparison theorem with the exponential function so I was not sure what I function...
  3. K

    Show the product of convergent sequences converge to the product of their limits

    Homework Statement Use the fact that a_n=a+(a_n-a) and b_n=b+(b_n-b) to establish the equality (a_n)(b_n)-ab=(a_n-a)(b_n)+b(a_n-a)+a(b_n-b) Then use this equality to give a different proof of part (d) of theorem 2.7. Homework Equations The theorem it is citing is: The sequence...
  4. C

    Calculus II - Alternating Series Test - Convergent?

    Hello! I was working some practice problems for a Calc II quiz for Friday on the alternating series test for convergence or divergence of a series. I ran into a problem when I was working the following series, trying to determine whether it was convergent or divergent: Homework Statement ∞...
  5. K

    Showing the sum of convergent and divergent sequence is divergent

    Homework Statement Show that the sum of a convrgent sequence and a divergent sequence must be a divergent sequence. What can you say about the sum of two divergent sequences? Homework Equations A theorem in the book states: Let {a_n} converge to a and {b_n} converge to b, then the...
  6. L

    Weakly convergent sequences are bounded

    Homework Statement I would like to show that a weakly convergent sequence is necessarily bounded. The Attempt at a Solution I would like to conclude that if I consider a sequence {Jx_k} in X''. Then for each x' in X' we have that \sup|Jx_k(x')| over all k is finite. I am not sure why...
  7. T

    Determine if integral is convergent or divergent?

    Homework Statement I attached the problem to this post. The Attempt at a Solution I was wondering if I could use the limit comparison test for this integral. My professor taught us this test that can be used for series but could it work for improper integrals as well? So what I would...
  8. S

    Convergent sequence property and proving divergence

    I feel like I'm missing something obvious, but anyway, in the text it states: lim as n→∞ of an+bn = ( lim as n→∞ of an ) + ( lim as n→∞ of bn ) But say an is 1/n and bn is n. Then the limit of the sum is n/n = 1, but the lim as n→∞ of bn doesn't exist and this property doesn't work...
  9. J

    Why is Harmonic Series Convergent

    I understand that the harmonic series, \frac{1}{x} is divergent because: \int (1/x) from one to infinity is: [ln(infinity) - ln(1)] which is clearly divergent. BUT When I look at the graph of \frac{1}{x} versus \frac{1}{x^{2}} they both look like they are converging to zero as x...
  10. M

    Prove: Sum ai bi Converges if Sum ai & Sum bi Non-negative

    Hi Can someone please help me to prove or give a counter example is sum ai and sum bi are convergent series with non-negative terms then sum aibi converges I believe that if it doesn't say "non-negative terms" then this wouldn't be true. Am I correct? Since each of two non-negative...
  11. N

    Finding the Sum of Convergent Series t^(-n^2) for n=1 to ∞

    what is the sum of the following series? I know it's convergent (using ratio test) but I'm not able to work it out :( S=t^(-1) + t^(-4)+t^(-9)...t^(k^2)...to ∞ where t>1 Thanks
  12. 8

    Metric spaces and convergent sequences

    Homework Statement let {xi} be a sequence of distinct elements in a metric space, and suppose that xi→x. Let f be a one-to-one map of the set of xis into itself. prove that f(xi)→x Homework Equations by convergence of xi, i know that for all ε>0, there exists some n0 such that if i≥n0...
  13. F

    Uniformly convergent sequence proof

    Homework Statement Let f_n(x) be a sequence of functions that converges uniformly to f(x) on the interval [0, 1]. Show that the sequence e^{f_n(x)} also converges uniformly to e^{f(x)} on [0,1]. Homework Equations The definition of uniform convergence. The Attempt at a Solution I...
  14. alexmahone

    MHB Give an example of a convergent series

    Give an example of a convergent series $\sum a_n$ for which $(a_n)^{1/n}\to 1$. PS: I think I got it: $\sum\frac{1}{n^2}$
  15. C

    Proving Convergence: Showing That x_n and y_n Have the Same Limit

    Homework Statement Show that if x_n is a convergent sequence, then the sequence given by that average values also converges to the same limit. y_n=\frac{x_1+x_2+x_3+...x_n}{n} The Attempt at a Solution Should I say that x_n converges to some number P. so now I need to show that y_n...
  16. N

    Determining of a sequence is convergent or divergence

    Homework Statement For x_{n} given by the following formula, establish either the convergence or divergence of the sequence X = (x_{n}) x_{n} := (-1)^{n}n/(n+1)Homework Equations The Attempt at a Solution This is for my real analysis class. I tried to use the squeeze theorem, but didn't get...
  17. Z

    The set of convergent subsequences

    Hello all, (*) I have a question about convergent subsequences. Specifically I am looking for an example of a sequence that is unbounded but who has convergent subsequences in the interval [0,1]. A similar question would be to have an unbounded sequence, but who has a convergent...
  18. Z

    Convergent Sequences and Functions

    Hello all, I am having trouble with a convergent series problem. The problem statement: Let f:ℝ→ℝ be a function such that there exists a constant 0<c<1 for which: |f(x)-f(y)| ≤c|x-y| for every x,y in ℝ. Prove that there exists a unique a in ℝ such that f(a) = a. There is a provided hint...
  19. M

    Finding sum to convergent series?

    Homework Statement Decide whether convergent or divergent, if convergent, find sum. Ʃ as n = 1 and goes to infinity > (3^n + 2^n)/6^n Homework Equations a/1-r The Attempt at a Solution I'm just confused where to find the "r" to this without actually plugging in values for...
  20. D

    Good dayQuestion: Determine whether the series is convergent or

    Good day.. Question: Determine whether the series is convergent or divergent: Series starts at n=1 and goes to infinity.. Of 2/(n*4throot(2n+2)) What I mean is.. 2/(n*(2n+2)^(1/4)) Can someone tell me which test to try? I can't get it in the form of a p-series.. so I think maybe the...
  21. H

    Determine whether the series is convergent or divergent

    Homework Statement I have to find whether the following is Convergent or Divergent ∑ from n = 1 to infinity 2 / n(2n + 2)^(1/4) Actually it's the fourth root, this is just easier to write. Homework Equations According to the front of the sheet it's a quiz on P-Series and Integral Test I'm...
  22. A

    A Simple Convergent Series

    can someone please explain how to get the result WolframAlpha gets at the following link: http://www.wolframalpha.com/input/?i=sum+%282n%2B1%29%2F%28n+%28n%2B1%29+%28n%2B2%29%29 the sum is (2n+1)/(n (n+1) (n+2)) from n=1 to inf. and the result is 5/4 . Thanks.
  23. K

    (Comparison Theorem) Why is x/(x^3+1) convergent on interval 0 to infinity?

    ∫0->∞ x/(x^3 + 1) dx. Use comparison theorem to determine whether the integral is convergent of divergent. Homework Equations None. The Attempt at a Solution ∫0->∞ x/(x^3 + 1) dx = ∫0->∞ x/(x^3) dx = ∫0->∞ 1/(x^2) dx From my class I learned that ∫1->∞ 1/(x^2) dx , is convergent But...
  24. K

    Partial sums for convergent series

    Is it possible to find a non-recursive formula for the partial sums of a convergent series?
  25. D

    Show That \{x_n\} is Convergent: Step-by-Step Guide

    Homework Statement The problem is longer but the part I'm stuck is to show that \{x_n\} is convergent (I thought showing it is Cauchy) if I know that for all \epsilon > 0 exists n_0 such that for all n \geq n_0 I have that |x_{n+1} - x_n| < \epsilon Homework Equations A...
  26. S

    Can a bounded subsequence have infinitely many convergent subsequences?

    I'm not sure if I am confusing myself or not, but a friend and I were trying to figure this out. Basically, I know that if a sequence is bounded, we are guaranteed at least one convergent subsequences. However, is it possible for a bounded sequence to have infinitely many of such subsequences?
  27. B

    How to prove a strictly diagonally dominant matrix is convergent

    If A and b are given, I know how to use the Jacobi's method to find out whether or not A is convergent. But how should I prove that "Jacobi's method is convergent if A is diagonally dominant" using just those given letters and symbols?
  28. J

    {Sn} is convergent -> {|Sn|} is convergent

    {Sn} is convergent ---> {|Sn|} is convergent Homework Statement I need to prove that if {sn} is convergent, then {|sn|} is convergent. Homework Equations sn is convergent if for some s and all ε > 0 there exists a positive integer N such that |sn - s| < ε whenever n ≥ N. The Attempt at a...
  29. W

    Is this series convergent or divergent

    Homework Statement Me and my friend are debating on wether the follow seris is convergent or divergent. The seris is the sum of (-1)^n-1 * ln(n)/n. Homework Equations p test and comparision tests. And alternating series test The Attempt at a Solution My approach to this problem...
  30. R

    Convergence and Divergence of the Sequence nsin(npi)

    I need to find out if this function is convergent or divergent when finding the limit to infiniti. nsin(npi) How do I solve this? Do I use the squeeze theorem or lhospital rule?
  31. H

    Proof on Sequences: Sum of a convergent and divergent diverges

    Homework Statement Prove if sequence a_{n} converges and sequence b_{n} diverges, then the sequence a_{n}+b_{n} also diverges. Homework Equations The Attempt at a Solution My professor recommended a proof by contradiction. That is, suppose a_{n}+b_{n} does converge. Then, for...
  32. H

    Convergent Series and Partial Sums

    Homework Statement Let \sum_{n=1} a_n and \sum_{n=1} b_n be convergent series. For each n \in \mathbb{N}, let c_{2n-1} = a_n and c_{2n} = b_n. Prove that \sum_{n=1} c_n converges. Homework Equations The Attempt at a Solution Not sure whether the following solution is...
  33. pairofstrings

    Is it necessary to have two sequences for convergence or divergence to occur?

    Homework Statement A bounded sequence need not be convergent Can you show me an illustration which shows a sequence that is convergent? I don't understand when if lim n ---> infinity sn = l, the sequence sn converges to l or {sn}. Now what is l ? My attempt or understanding of the...
  34. D

    A convergent sequence of reals

    Call {a1, a2, a3, ...} = {an} a "convergent sequence" if \exists L \in \mathbb{R} : \quad \forall \epsilon > 0 \quad \exists N \in \mathbb{N} : (\forall n > N \quad (n > N \implies |a_n - L| < \epsilon)) in which case we write \lim_{n \rightarrow \infty} a_n = \lim a_n = L. Of course this...
  35. C

    Convergent series with non-negative terms, a counter-example with negative terms

    Homework Statement The terms of convergent series \sum_{n=1}^\inftya_n are non-negative. Let m_n = max{a_n, a_{n+1}}, n = 1,2,... Prove that \sum_{n=1}^\inftym_n converges. Show with a counter-example that the claim above doesn't necessarily hold if the assumption a_n\geq0 for all...
  36. A

    Bounded sequences and convergent subsequences in metric spaces

    Suppose we're in a general normed space, and we're considering a sequence \{x_n\} which is bounded in norm: \|x_n\| \leq M for some M > 0. Do we know that \{x_n\} has a convergent subsequence? Why or why not? I know this is true in \mathbb R^n, but is it true in an arbitrary normed space? In...
  37. R

    Convergent sum has possible error that never occurs

    I have a convergent sum where I use the reciprocal of a_n at each step: a_n = a_n / gcd(a_n, b_n) <--- I'm removing common factors. This converges as long as I want to run it. Both a_n and b_n are quite dynamic. However, if a_n equals b_n then the divide after the gcd would return 1...
  38. R

    Convergent subsequences in compact spaces

    My quick question is this: I know it's true that any sequence in a compact metric space has a convergent subsequence (ie metric spaces are sequentially compact). Also, any arbitrary compact topological space is limit point compact, ie every (infinite) sequence has a limit point. So in general...
  39. D

    Proving Convergence of \{b_n\} when \{a_n\}\to A, \{a_nb_n\} Converge

    If \{a_n\}\to A, \ \{a_nb_n\} converge, and A\neq 0, then prove \{b_n\} converges. Let \epsilon>0. Then \exists N_1,N_2\in\mathbb{N}, \ n\geq N_1,N_2 |a_n-A|<\frac{\epsilon}{2} And let \{a_nb_n\}\to AB So, |a_nb_n-AB|<\epsilon I don't know how to show b_n is < epsilon.
  40. E

    Prove that integral is convergent

    Homework Statement i need to know the integral of x^alpha times (lnx)^ beta from 0 to 0.5 the question is if alpha greater than -1 prove that integral convergent Homework Equations The Attempt at a Solution
  41. P

    Is this convergent? sum (sin k)

    \sum sin k and k is from 1 to infinity,thx
  42. J

    Prove that a sequence of functions has a convergent subsequence

    Homework Statement Let \{ f_{n} \}_{n=1}^{\infty} \subset C[0,1] be twice differentiable, and satisfying 0 = f_{n}(0) = f'_{n}(0) and \| f''_{n}\|_{\infty } . Prove that \{ f_{n} \}_{n=1}^{\infty} has a convergent subsequence. Homework Equations So since C[0,1] is a compact metric...
  43. S

    Proof: every convergent sequence is bounded

    Homework Statement Prove that every convergent sequence is bounded. Homework Equations Definition of \lim_{n \to +\infty} a_n = L \forall \epsilon > 0, \exists k \in \mathbb{R} \; s.t \; \forall n \in \mathbb{N}, n \geq k, \; |a_n - L| < \epsilon Definition of a bounded sequence: A...
  44. S

    Overdamping vs. convergent oscillation

    I had a question regarding oscillatory motion in a spring-mass-damper system. I understand the concepts of over, under, and critical damping and the criteria which determine them, but I'm wondering if they are equivalent to the concepts of convergent, divergent, and stable oscillation. I...
  45. D

    How can I find a starting point for problems with convergent subsequences?

    I am having a hard time finding a starting point for these problems. One is to find a sequence with subsequences that converge to 1, 2, and 3. A similar problem (which would solve both problems) is to find a sequence that has subsequences that converge to every positive integer. I am not...
  46. T

    Convergent and Divergent Series

    Homework Statement For what integer k, k > 1, will both sigma n=1 to infinity ((-1)^(kn))/n and sigma n=1 to infinity (k/4)^n converge? A) 6 B) 5 C)4 D)3 E)2 Homework Equations The Attempt at a Solution I tried to use the ratio test and after some simplifying I got (-1)^k (n/n+1)...
  47. P

    How to determine if the series is convergent or divergent.

    Homework Statement Determine if the series is convergent or divergent. \sum x^2e^{-x^2} Homework Equations The Attempt at a Solution x^2e^{-x^2}=\frac{x^2}{e^{x^2}} \lim_{x\to\infty } \frac{(x+1)^2}{e^{(x+1)^2}}\frac{e^{x^2}}{x^2} and since (x+1)^2=x^2+2n+1 and...
  48. Z

    Does this real sequence necessarily converge?

    Homework Statement Let (x_n) be a real sequence which satisfies |x_n - x_(n+1)| < (1/n) for all natural numbers n. Does (x_n) necessarily converge? Prove or provide counterexample. Homework Equations Cauchy Criterion for sequences The Attempt at a Solution I figured at first...
  49. Char. Limit

    Comparing Sets of Convergent Sequences and Series

    So I had this question in PF chat, but I decided this would be a better place for it. Say I have two sets, S and S'. S is the set of all convergent sequences. S' is the set of all convergent series...es. Is S larger than S', and if so, how much larger?
  50. J

    Proving Convergence of sum[n=0 to inf] (z+2)^(n-1)/((n+1)^3 * 4^n) for |z+2|<=4

    Homework Statement Prove that the series sum[n=0 to inf] (z+2)^(n-1)/((n+1)^3 * 4^n) converges for |z+2| <=4 Homework Equations The Attempt at a Solution sum[n=0 to inf] (z+2)^(n-1)/((n+1)^3 * 4^n) <= sum[n=0 to inf] |(z+2)^(n-1)/((n+1)^3 * 4^n)| <= sum[n=0 to inf]...
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