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. T

    Sum of convergent and divergent series

    Homework Statement Prove that if $\displaystyle\sum_{n=1}^\infty a_n$ converges and $\displaystyle\sum_{n=1}^\infty b_n$ diverges, then $\displaystyle\sum_{n=1}^\infty (a_n+b_n)$ diverges. Homework Equations I know that the limit of {a_n} = 0 because it is convergent, but I can't say...
  2. D

    Proving Convergence of Absolute Value Sequence in Real Numbers

    For a sequence in the reals {an} converges to a, show {|an|} converges to |a|. For any e>0 the exists an N s.t. for any n>N |an-a|<e I want to use this inequality, but there is something funny going on. I do not know how to justify it. |an-a|\leq||an|-|a||
  3. J

    What is the Convergent Series Sum Formula?

    Homework Statement Dear All, I have a series that I know to converge but for which I can't work out the infinite sum. It should be something simple. S_n = \sum_{j=1}^\infty \beta^j j Can somebody help me with this? I think the solution is: \frac{\beta}{(1-\beta)^2}
  4. R

    Showing a sequence is bounded and convergent to its infimum.

    Homework Statement Show that any non-increasing bounded from below sequence is convergent to its infimum. Homework Equations Not quite sure... is this a monotonic sequence? The Attempt at a Solution At this point I'm not even sure about which route to go. I am in need of...
  5. G

    How to show a series of functions is not uniformly convergent

    Ok, I chose to ask about ways to determine if a series of functions is NOT uniformly convergent because I think that would best answer the overall difficulties I have with uniform convergence. I have a good idea of what uniform convergence is, I can give the definition, and if the problem is...
  6. J

    Convergent Filter Base and Continuous Function

    Hi All, I can't see how the following is proved. Given two topological space (X, T), (Y, U) and a function f from X to Y and the following two statements. 1. f is continuous, i.e. for every open set U in U, the inverse image f-1(U) is in T 2. For every convergent filter base F -> x, the...
  7. S

    Why the sequence is not convergent?

    In a book I am reading, they mention the following as an example of a Cauchy sequence which is not convergent: Consider the set of all bounded continuous real functions defined on the closed unit interval, and let the metric of the set be d(f,g)=\int_0^1 \! |f(x)-g(x)| \, dx. Let (f_n) be a...
  8. J

    Is the Convergence of this Infinite Series Dependent on the Value of Alpha?

    Homework Statement For what values of alpha is the following series convergent: \sum_{v=1}^{\infty} \frac{(-1)^{n-1}}{n^{\alpha}} = 1 - \frac{1}{2^{\alpha}} + \frac{1}{3^{\alpha}} + ... Homework Equations The Attempt at a Solution For negative alpha and alpha = 0 the series...
  9. W

    (2) questions on convergent and divergent

    1. A year after the leak began the chemical had spread 1500 meters from its source. After two years, the chemical had spread 900 meters more, and by the end of the third year, it had reached an additional 540 meters. a. If this pattern continues, how far will the spill have...
  10. E

    Convergent non-monotone sequences

    Homework Statement Let C=\bigcupn=1\inftyCn where Cn=[1/n,3-(1/n)] a) Find C in its simplest form. b)Give a non-monotone sequence in C converging to 0. Homework Equations The Attempt at a Solution For part a) i get C=[0,3]. Is this correct? I am not sure as to wether 0 and 3 are...
  11. G

    Sum of convergent series HELP!

    Sum of convergent series HELP! Homework Statement Find the sum of the convergent series - the sum of 6 / (n+7)(n+9) from n=1 to infinity (∞) A) 31/24 B) 45/56 C) 8/11 D) 17/24 E) 23/24 2. The attempt at a solution I was looking in the book and they had one example that was kinda...
  12. J

    Conditionally convergence of a product of a conditonally convergent series

    Homework Statement 1. Find a function such that infinite sum of anis conditionally convergent but infinite sum of (an)^3 does not converge conditionally The Attempt at a Solution We have confusions about the definition of conditional convergence. Importantly, does absolute convergence also mean...
  13. M

    Find c likes the series is convergent

    Find c ∈ IR, like ∫∞ ( 2x - c ) dx is convergent 0 X^2 +1 2x+1 I need your help because I was trying to resolve the problem, but I couldn't, is difficult for me. Please help me!
  14. F

    Is this series convergent or divergent?

    convergent or divergent?? Homework Statement i took a calc 2 quiz today and had a question on one of the. it's too late to correct what i did, but it's never too late to learn it for the final haha here is the problem they want us to find the sum of the series (below) from 1 to...
  15. P

    Find the sum of the following convergent series

    Sorry about the title, if possible please change it 1. Find the sum of the following convergent series \sum_{j=0}^{\infty}(-1)^{j}(2/3)^{j} 2. \sum_{j=0}^{\infty}c^{j} = 1/(1-c) if |c| < 1 The Attempt at a Solution\sum_{j=0}^{\infty}(-1)^{j}(2/3)^{j} = 1 - 2/3 + (2/3)^{2} + ... = 1 - (2/3 +...
  16. T

    Is the Series Convergent or Divergent?

    Homework Statement Determine whether the series converges or diverges. \sum 3+7n / 6n Attempt : Comparison test : 3+7n / 6n < 7n / 6n 3+7n / 6n < (6/7)n since (6/7)n is a geometric series and is convergent is 3+7n / 6n convergent as well?
  17. T

    Another convergent and divergent

    Homework Statement Determine whether the series is convergent or divergent. \sum n5 / (n6 + 1)
  18. T

    Determine whether the series is convergent or divergent

    Homework Statement Determine whether the series is convergent or divergent. 1 + 1/8 + 1/27 + 1/64 + 1/125 ... Homework Equations The Attempt at a Solution I know this is convergent but not sure how to prove this mathematically.
  19. Q

    Polynomial sequence uniformly convergent on annulus

    Homework Statement Can we find a sequence, say p_j(z) such that p_j ---> 1/z uniformly for z is an element of an annulus between 1 and 2, that is 1 < abs(z) < 2? Then i am asked to do the same thing but for p_j ---> sin(1/z^2).Homework Equations Not too sure about this, maybe Taylor...
  20. T

    Limit of a convergent series and a divergent sequence

    Homework Statement Show that if: lim_{k\to\infty}b_k\to+\infty, \sum_{k=1}^\infty a_k converges and, \sum_{k=1}^\infty a_k b_k converges, then lim_{m\to\infty} b_m \sum_{k=m}^\infty a_k = 0 Homework Equations The Attempt at a Solution I only have an idea why this is true--\sum a_k...
  21. M

    Sum of the convergent infinite series ln(n)/n^2

    Homework Statement Find the sum of the series: ln(n)/n^2 from n=1 to infinity. I already know that it is convergent(at least i hope i am right on that fact) Homework Equations The Attempt at a Solution I tried to use geometric series but i can't see anything like that that would...
  22. I

    Convergent or Divegent Series?

    Homework Statement \sum{(ln(k))/(\sqrt{k+2})}, with k starting at 1 and going to \infty Homework Equations Does this series converge or diverge? Be sure to explain what tests were used and why they are applicable. The Attempt at a Solution Okay, my TA got that this diverges, but I...
  23. E

    Convergent series. Is my logic correct?

    Show \sum_1^\infty\frac{x^n}{1+x^n} converges when x is in [0,1) \sum_1^\infty\frac{x^n}{1+x^n} = \sum_1^\infty\frac{1}{1+x^n} * x^n <= \sum_1^\infty\frac{1}{1} * x^n = \sum_1^\infty x^n The last sum is g-series, converges since r = x < 1
  24. I

    Proving Uniform Convergence of f_n = x/sqrt(1+nx^2) on R

    Homework Statement Prove that f_{n} = \frac{x}{\sqrt{1+nx^2}} is uniformly convergent to 0 on all real numbers Homework Equations {f_n} is said to converge uniformly on E if there is a function f:E->R such that for every epsilon >0, there is an N where n>=N implies that | f_n(x) - f(x) |...
  25. I

    The product of absconverg series and bounded seq is absolutely convergent

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  26. I

    A weird convergent series involving integral

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  27. F

    Indefinite Integral of an Absolute Convergent Function

    Hi, I was wondering if a function is absolutely convergent over a certain interval, say, (0,\infty) will its indefinite integral also be absolutely convergent over the same interval? Also, assume that f(x) is convergent for (0,\infty). Would g(x) = \int{\int_{0}^{\infty}f(x)dx}dy &=&...
  28. B

    Is the exponential series uniformly convergent?

    Is the series \sum_{n=0}^{\infty}\frac{z^n}{n!} uniformly convergent for all z in the complex plane? It is uniformly convergent for all z in any bounded set, but the complex plane is unbounded. My instinct is that it is NOT uniformly convergent for all z in C. This is not homework.
  29. S

    Convergent Series Proof

    Homework Statement If the sequence of partial sums of |a_n| is convergent and b_n is bounded, prove that the sequence of partial sums of the product (a_n)(b_n) is also convergent. Homework Equations Cauchy sequences and bounded sequences The Attempt at a Solution I wrote the...
  30. F

    Convergent or Divergent?

    If a power series, \sumc(subk)*x^{k} diverges at x=-2, then it diverges at x=-3. True or False? I said true, but was confused by my reasoning. Does anyone have any suggestions?
  31. Somefantastik

    Convergent Integrals: Best Place to Start?

    When showing that an integral is convergent, where is the best place to start?
  32. P

    Easy convergent subsequence question.

    Homework Statement Consider the sequence {x_k} = {(arctan(k^2+1),sink)} in R^2. Is there a convergent subsequence? Justify your answer. Homework Equations Every bounded sequence in R^n has a convergent subsequence. The Attempt at a Solution To show {x_k} is bounded: The range...
  33. B

    Proving a Sequence is Convergent

    Homework Statement Let {a_{n}}^{\infty}_{n=1} be a sequence of real numbers that satisfies |a_{n+1} - a_{n}| \leq \frac{1}{2}|a_{n} - a_{n-1}| for all n\geq2 Homework Equations The Attempt at a Solution So, I know that it suffices to show that the sequence is...
  34. G

    Give an example of such a convergent series

    Homework Statement Give an example of a convergent series \Sigma z_{n} So that for each n in N we have: limsup abs{\frac{z_{n+1}}{z_{n}}} is greater than 1
  35. A

    Is (n!/n^n)*exp^n a Convergent or Divergent Summation?

    1)summation for n=0 to infinity for (n!/n^n)*exp^n. Can anyone help to prove whether this is convergent or divergent?
  36. D

    Proving convergent for the products of two series.

    Homework Statement I'm been trying to wrap my head around this one for a couple of days. I have a problem that states the following. (All sums go from n to \infty" Suppose we know that \sumAn and \sum B n are both convergent series each with positive terms. Show that the series \sumAn Bn must...
  37. N

    Determine whether series is convergent or divergent

    Homework Statement Determine whether or not the series \sum^{\infty}_{n=1} \frac{1}{\sqrt{n+1}+\sqrt{n}} converges. The Attempt at a Solution Assuming this diverges, I rationalize it to get get \sum^{\infty}_{n=1} \sqrt{n+1} - \sqrt{n}. How would I proceed further? Is this even the...
  38. N

    Two identical convergent lenses

    Homework Statement the figure shows a combination of two identical lenses. There is a 2cm tall object that is 36 cm away from the first lens (f=9 cm). The second lens is 15 cm away from the first lens. It also has a focal length of 9 cm. So, the first question was "find the position of the...
  39. K

    Convergent Sequences on l infinity

    Homework Statement Define R^\infty_f = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \exists k_0 \text{ such that } t^{(k})=0 \; \forall k\geq k_0 \} Define l^\infty = \{ (t^{(1}),t^{(2}), \ldots, ) |\; t^{(i}) \in \mathbb{R}\; \forall i, \; \sup_{k\geq 1} |...
  40. D

    Convergent Series Identities

    [SOLVED] Convergent Series Identities Homework Statement a) If c is a number and \sum a_{n} from n=1 to infinity is convergent to L, show that \sum ca_{n} from n=1 to infinity is convergent to cL, using the precise definition of a sequence. b)If \sum a_{n} from n=1 to infinity and \sum...
  41. K

    Convergent Sequences: Prove Lim d(x_n,y_n)=d(a,b)

    Homework Statement Let (X,d) be a metric space with two sequences (x_n), (y_n) which converge to values of a,b respectively. Show that \lim_{n \to \infty} d(x_n,y_n) = d(a,b) Homework Equations (x_n) \rightarrow a \Leftrightarrow \forall \epsilon >0 \quad \exists n_0 \in...
  42. A

    Uniform Convergence of \{\frac{n^2x}{1+n^3x}\} on Different Intervals

    I need to determine whether the sequence \{\frac{n^2x}{1+n^3x}\} is uniformly convergent on the intervals: [1,2] [a,inf), a>0 For the first one, I notoced the function is decreasing on the interval, so the \sup|\frac{n^2x}{1+n^3x}| will be when x=1, and when x=1, the sequence goes to 0...
  43. B

    How Can You Find the Sum of a Convergent Series with Partial Fractions?

    Homework Statement Find the sum of the convergent series: The sum of 1/ (n^2 - 1) from n=2 to infinity Homework Equations The Attempt at a Solution I want to break it down into 2 fractions and use partial fractions. 1/(n-1)(n+1)...but I don't know where to go from here...
  44. B

    Unravelling the Mystery of Cn: How to Find Convergent Sequences

    Sequences HELP! Homework Statement Show that the sequence Cn = [(-1)^n * 1/n!] Homework Equations The Attempt at a Solution This is an example in my book but I am not understanding it... It says to find 2 convergent sequences that can be related to the given sequence. 2 possibilities are...
  45. P

    Divergent Harmonic Series, Convergent P-Series (Cauchy sequences)

    Homework Statement (a) Show that \sum \frac 1n is not convergent by showing that the partial sums are not a Cauchy sequence (b) Show that \sum \frac 1{n^2} is convergent by showing that the partial sums form a Cauchy sequenceHomework Equations Given epsilon>0, a sequence is Cauchy if there...
  46. S

    Is My Thinking Wrong About Convergent or Divergent?

    For lim n->infinity n^-(1+1/n), the p series test shows that it converges since (1+1/n) will be greater than 1, while doing a limit comparison test with 1/n gives 1 showing that it diverges since 1/n diverges. For which one is my thinking wrong about?
  47. N

    Series- convergent or divergent?

    from n=1 to infinity does the series converge or diverge? n!/n^n its in the secition of the book with the comparison test and limit comparison test. if you compare it with 1/n^n (this is a geomoetric series) you get a= 1/n amd r= 1/n but in the thrm r = to some finite number...
  48. A

    Unique limit of a convergent filter

    Question: Prove that if any convergent filter on a space X converges to a unique limit, then X is Hausdorff. I think the solution in my textbook is faulty. It says "Suppose X is not Hausdorff. Let Nx and Ny be the collection of all neighbourhoods of x and y, respectively. Then Nx U Ny...
  49. D

    Convergent and Divergent problem

    If I have (a_n + b_n)^n = c_n where a_n is convergent and b_n divergent. Is c_n then divergent? And what if a_n and b_n were divergent, would c_n be divergent also? but what if they were both convergent then surely c_n is convergent right? I can't see a rule or a theorem that tells me...
  50. D

    Test Series Convergence: 1! + 2! + ... + n! / (2n)!

    Hi, could you please check if my solution is correct? Homework Statement Test the following series for convergence: \sum_{n=1}^{\infty}\frac{1!+2!+...+n!}{(\left 2n \right)!} The Attempt at a Solution I can use a slightly altered series \sum_{n=1}^{\infty}\frac{nn!}{(\left 2n \right)!}...
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