What is Comparison: Definition and 388 Discussions

Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are similar to the other, which are different, and to what degree. Where characteristics are different, the differences may then be evaluated to determine which thing is best suited for a particular purpose. The description of similarities and differences found between the two things is also called a comparison. Comparison can take many distinct forms, varying by field:

To compare is to bring two or more things together (physically or in contemplation) and to examine them systematically, identifying similarities and differences among them. Comparison has a different meaning within each framework of study. Any exploration of the similarities or differences of two or more units is a comparison. In the most limited sense, it consists of comparing two units isolated from each other.
To compare things, they must have characteristics that are similar enough in relevant ways to merit comparison. If two things are too different to compare in a useful way, an attempt to compare them is colloquially referred to in English as "comparing apples and oranges." Comparison is widely used in society, in science and in the arts.

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  1. A

    Series Comparison Test, help?

    Series Comparison Test, URGENT help? Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for "correct") if the argument is valid, or enter I (for...
  2. L

    Binomial expansion comparison with legendre polynomial expansion

    Hi, I've been working on this question which asks to show that {{P}_{n}}(x)=\frac{1}{{{2}^{n}}n!}\frac{{{d}^{n}}}{d{{x}^{n}}}{{\left( {{x}^{2}}-1 \right)}^{n}} So first taking the n derivatives of the binomial expansions of (x2-1)n...
  3. 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...
  4. H

    Using the limit comparison test

    I'm given the following: 3/(n(2^(n-1))) I have to determine convergence using the limit comparison test. I've proved its convergent using the ratio test but am struggling with which term do I divide the above for the limit comparison test. Help?
  5. C

    Comparison of strong nuclear force and electromagnetic force

    How much stronger is the strong nuclear force compared it to the electromagnetic force beyond what could be accounted for by the inverse of the distance squared?
  6. O

    Finding divergence/convergence by direct comparison test

    Homework Statement \sum^{∞}_{1}1/n^{n} Homework Equations Direct comparison test The Attempt at a Solution Since the main factor in the equation is the exponent that would be changing as n goes to infinity, I know that from the p series as p > 1 the the series converges. So I know...
  7. Square1

    Comparing Stewart's Integration Strategy: Proving g Convergence/Divergence

    Just as talked about in stewart in strategy for integration. I found notes online that also say: g(x) >= f(x) >= 0, then you want to prove convergence on g. If f(x) >= g(x) >= 0, then you want to prove divergence on g. I am pretty sure I follow the logic here, but how exactly does one pick...
  8. T

    Series Comparison Test for Sum of (1/(sqrt(n^2+1))) and (1/(2n))

    Homework Statement For the sum from n=1 to ∞ (1/(sqrt(n^2+1)), I know you can use the limit comparison test to show that it is divergent but I was wondering if it is possible to compare this with 1/(2n)? I am not sure if 1/(2n) is always less than (1/(sqrt(n^2+1)) within those bounds. How...
  9. D

    Deciding if Divergent/Convergent via Comparison Theorem

    Homework Statement Use the Comparison Theorem to determine whether the integral is convergent or divergent. \int_{0}^{1}\frac{e^{-x}}{\sqrt{x}} Homework Equations The Attempt at a Solution \int_{0}^{1}\frac{e^{-x}}{\sqrt{x}} \leq \int_{0}^{1}\frac{1}{x^{\frac{1}{2}}} Because p<1, the...
  10. A

    Comparison of Net Electric Force on a +q

    Homework Statement In cases A and B shown there are two positive charges +Q each a distance s away from a third positive point charge +q. Is the net electric force on the +q charge in case A greater than, less than, or equal to the net electric force on the +q charge in case B? Explain...
  11. S

    Comparison test for convergence problem: why is this incorrect?

    Homework Statement The original question is posted on my online-assignment. It asks the following: Determine whether the following series converges or diverges: \sum^{\infty}_{n=1}\frac{3^{n}}{3+7^{n}} There are 3 entry fields for this question. One right next to the series above...
  12. T

    Use comparison theorem to determine if \int_{0}^{1}\frac{e^{-x}}{\sqrt{x}}~dx

    Homework Statement The question says use the comparison to determine if ##\int_{0}^{1}\frac{e^{-x}}{\sqrt{x}}dx## converges. What should I compare to? Homework EquationsThe Attempt at a Solution
  13. O

    Wind Power Generation Comparison

    Homework Statement Two sites are being considered for wind power generation. In the first site, the wind blows steadily at 7 m/s for 3000 hours per year, whereas the second site the wind blows at 10m/s for 2000 hours per year. Assuming wind velocity is negligible at other times for simplicity...
  14. D

    Bond angle comparison in sp2 hybridised carbon atoms

    1. Homework Statement This was a question on an exam I took a week ago in inorganic chemistry. The problem statement was like: "We have two molecules, H2CCH2 (ethene) and F2CCF2 (1,1,2,2-tetrafluoroethene). Which bond angle is greater? H-C-H in ethene or F-C-F in 1,1,2,2-tetrafluoroethene?"...
  15. T

    Regression - AIC/SBC Comparison

    I'm not sure if this is the right place for this question, but it was on the comparison between different model's AIC/SBC values. I ran a linear regression and got an AIC/SBC of .743/.768. When I ran the same regression in log-linear form I ended up with an AIC/SBC of -7.559/-7.534. My...
  16. R

    How much faster does time pass in space in comparison to that on Earth?

    The title says it all really. Thanks.
  17. H

    Matter & Photon Particle Comparison

    Homework Statement Compare and contrast a 2.2 eV photonn with a 2.2 eV electron in terms of energy(J), rest mass (Kg) speed (m/s) wavelength (m) and momentum kgm/s. Homework Equations For Photon P=E/C F=E/h V=fλ λ= h/P For Electron λ=h/mv m=E/C2 p=h/λ Ke=1/2mv2 The...
  18. C

    Show convergence using comparison test on sin(1/n)

    Homework Statement a) Test the following series for convergence using the comparison test : sin(1/n) Explain your conclusion. Homework Equations The Attempt at a Solution i must show f(x)<g(x) in order for it to converge other wise divergence. g(x) = 1/n sin(1/n) >...
  19. H

    Which is a Better Approximation: (1+x)^n or e^nx? How to Show?

    This isn't a coursework problem. I'm on winter break. Homework Statement A common approximation used in physics is: (1+x)n ≈ 1+nx for small x This implies that lim(x→0) (1+x)n = lim(x→0) 1+nx which is a true statement. However, lim(x→0) (1+x)n = lim(x→0) [(1+x)1/x]xn = lim(x→0) exn This...
  20. R

    Quantitive Histogram Comparison

    I have two histograms that I would like to compare quantitatively. The values of the first histogram have respective relative errors for each bin. The second histogram has no statistical uncertainty. I could compute probabilities for each bin that the exact values would fall into a given...
  21. T

    Comparison of Riemann integral to accumulation function

    Let f:[0,1]→ℝ be an increasing function. Show that for all x in (0,1], \frac{1}{x}\int_{0}^{x}f (t) \,dt \le \int_{0}^{1}f (t) \,dt So by working backwards I got to trying to show that (1-x)\int_{0}^{1}f (t) \,dt \le \int_{x}^{1}f (t) \,dt . While I know both sides are equal at x=1, the...
  22. M

    Comparison test to determine convergence

    Use a comparison test to determine whether the series \sum (n+1)/(n^{2}+n+1) diverges or converges. I started out by simplifying the series to 1/n+1 and then from there I compared it to 1/n, which converges. 1/n is greater than 1/n+1 so based on the comparison test, the original series...
  23. F

    Kapteyn's and Shapley's Milky Way - a comparison

    Hi all, new here and was wondering if you could help me out. I'm basically giving the two Milky Way models, and what was a common mistake (note it's a written piece not workings out) and I'd like to know if you think it makes sense/I don't waffle too much. Thanks in advanceThe attempt at a...
  24. J

    Comparison Test Problem & Estimitaing it's error

    Homework Statement The question in the book is: "Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. My problem is estimating the error I'm looking for. I just need help with finding the integral. Homework Equations ((sin n)^2)/(n^3) The...
  25. T

    Comparison Test: Am I using a good comparison function?

    Homework Statement Does the following interval diverge? \int^9_1 \frac{-4}{\sqrt[3]{x-9}} Homework Equations The Attempt at a Solution Well.. I've used the following function that I think is always less than the above function to prove that the function above DOES NOT diverge (by...
  26. J

    Does This Series Converge or Diverge?

    Homework Statement Determine whether the series converges or diverges. What I would like is some type of information on how to continue the problem. Homework Equations ∞ Ʃ √(n+1)/2n^2+n+1 n=1 The Attempt at a Solution I was thinking of doing a comparison test by doing...
  27. P

    Convergence of \sum_{n=1}^\infty \frac{1}{n!}: A Basic Comparison Test

    \sum_{n=1}^\infty \frac{1}{n!} I understand what n! means, but I have no clue what to compare this to. It is obvious to me that the sum converges, but I'm not sure how to prove it. I assume I would compare it to a p-series but I need help. Thanks!
  28. A

    Mutual Exclusiveness vs Independence: A Comparison

    what is the basic difference between mutual exclusiveness and independence? actually i got this as a difference between fundamental principle of addition and fundamental principle of multiplication. it is quite urgent.
  29. N

    Physics II in comparison with Physics I?

    Alright here is my story: I took high school physics and loved it, went to college and took a class that is basically an algebra based physics class and a very light introductory just because I couldn't wait to get my hands on physics. When I was able to take Calculus based physics I did-- but...
  30. S

    Q: A&B moving in opposite directions @ 0.6 c comparison with sound.

    I'm trying to understand relativity on a simple example: 2 objects A & B moving in exactly opposite directions at 0.6c each, starting from frame C at the center of things that remains stationary. So, we have: W <----0.6c A ...... C ..... B 0.6c----> E 1. From frame C, A is moving...
  31. M

    Can the Limit Comparison Test Determine the Convergence of this Series?

    Homework Statement \sum_{n=2}^{\infty}\frac{1}{n\sqrt{n^2-1}} Homework Equations direct comparison test limit comparison test The Attempt at a Solution so i kind of cheated and looked at the back of my book and it says to compare with \frac{1}{n^{3/2}} so i tried using the direct...
  32. B

    Comparison of width of a wavefunction in real space and momentum space

    Hello, I have a slight problem with Quantumtheory here. Homework Statement I have solved the schrödinger equation in the momentum space for a delta potential and also transferred it into real space. So now I have to find the correlation between the width of the wavefunction in both spaces...
  33. L

    Orbital Transfer DeltaV Comparison for Different Orbits

    Homework Statement Assuming planar orbits, calculate the Hohmann ∆V required to transfer from a low-Earth circular orbit with radius of 1.03 Earth radii to the orbit of the Moon (assume 60 Earth radii). Compare this to the 3-maneuver strategy of (1) escaping to infinity on a parabolic orbit...
  34. S

    Monatomic Linear Chain - Comparison of Numerical and Analytical Results

    Hi all, I'm having a few problems with crystal dynamics of a simple monatomic chain. Taking the dispersion relation: \omega^2 = \frac{4k}{m}\left(\sin^2 \left( \frac{\kappa a}{2}\right)\right) Where k=spring constant, m=mass, \kappa=wavevector, a= lattice constant and \omega=...
  35. P

    Comparison test for basic integral

    integral 1/(1+x^2)dx from 0 to infinity I decided to compare that to 1/(1+x) (saying 1/(1+x) > (1/(1+x^2)) but this diverges when the original equation converges. Can someone explain why the integral 1/(1+x) was not a proper choice and what the process would be to find a correct comparison.
  36. P

    Proper use of comparison theorum?

    \int_{0}^{\frac{pi}{2}} \frac{cos(x)}{x^{1/2}} dx \int_{0}^{\frac{pi}{2}} \frac{cos(x)}{x^{1/2}} dx \le \int_{0}^{\frac{pi}{2}} \frac{1}{x^{1/2}} dx This would mean that the original equation was convergent. Was my reasoning correct in making the numerator 1 since the max value of cos(x) is...
  37. P

    Router's IP vs. LAN IP: A Comparison

    what is the difference between router's ip address and my lan ip adress... what i understood is that we have a lan ip adress and a wan ip adrress ...now my wan ip address is found using what's my ip address address.com and it changes...hence dynamic...and my lan ip address is using...
  38. B

    Improper Integral Comparison Proof

    Homework Statement Prove or disprove: b\int_b^∞ f(x) dx ≤ \int_b^∞ xf(x) dx for any b≥0 and f(x)≥0Homework Equations N/A The Attempt at a Solution Ok this question has caused me quite some problems. I have come to the conclusion that this needs to be proven rather than disproven...
  39. S

    Direct comparison test, just need some explanation

    2 problems, I need to use the direct comparison test with either a p series or a geometric series 1)series of j^2/(j^3 +4j +3) I thought of comparing it to j^2/J^3 which comes out to 1/j, but that dosent work, my teachers answer is you compare it to 1/5j 2) series of sqrt(q)/(q+2) I would...
  40. P

    [KL-divergence] comparison of pdf's

    Hi all, I am trying to devise a mathematical model for my project I am working at. Description is as follows: we have a sample space \Omega=\{w_1,w_2,\cdots, w_N\} It is very large. Suppose further, that we have some assumption of frequency of occurrence of each w_i , stored in...
  41. icesalmon

    Stuck with a limit comparison test problem.

    Homework Statement Determine the Convergence or Divergence of \sum_{n=1}^\infty\left(sin(1/n)\right) Homework Equations limit comparison test The Attempt at a Solution I don't know what to compare this series to, I tried the harmonic series to get sin(1/n)/(1/n) = nsin(1/n) which...
  42. W

    Comparing Series Convergence: Limit or Direct Comparison Test?

    Homework Statement Does the following series converge or diverge (use either the Limit Comparison or the Direct Comparison Test): \sum_{n=1}^{+\infty} \frac{3^{n-1}+1}{3^{n}} Homework Equations In a previous problem that was \sum_{n=1}^{+\infty} \frac{1}{3^{n-1}+1} I was able to...
  43. W

    Light Bending: Comparison of Theories

    In the essay „Über die Ablenkung des Lichtes I am Schwere*feld der Sonne“ ( http://adsabs.harvard.edu/full/1931ZA...3..171F ) the authors - Freundlich, Klüber and Brunn - presented 1931 graphically the results of three expeditions, which took place 1919, 1922 and 1929. They put together all...
  44. M

    Need help with comparison method for improper integral?

    Homework Statement \int \frac{arctan(x)}{2+e^x}dx where the interval of the integrand is from 0 to infinity. In order to use the comparison method I need to compare 2 functions but I am having so much difficulty figuring out what function to compare it to. Its not just this particular...
  45. T

    Solving Airy's Equation and Applying the Sturm Comparison Theorem

    Homework Statement (a) By using a suitable transformation, show that the normal form of the DE y'' - 2y' + (x+1)y = 0\;\;\;\;\;(*) is Airy's equation u'' + xu = 0. (b) State the Sturm comparison theorem for zeros of 2 second order linear DEs in normal form. (c) By comparing with the DE...
  46. C

    Series comparison test question

    Homework Statement Summation from n=1 to infinity: (e^(1/n)) / n Homework Equations The Attempt at a Solution Can someone point out what criteria I should be considering when trying to determine which test to use? I was looking at a comparison test as a way to go on this one...
  47. C

    Series comparison test question

    Problem: Summation from n=1 to infinity: 1/(n^3 + n^2) I understand, for example, another problem wherein it is 3/(4^n + 5) what you would compare that one to but how do you go about breaking this one up with two "n" terms? Are you supposed to, in general, pick the largest value of n? So...
  48. M

    Series, Comparison and Limit Tests

    Im practicing problems for my Calc 2 test tomorrow. I am doing this problem which I am not quite sure I've done it right. I think I have, but I want confirmation... \sum_{n=1}^{\infty} \frac{1}{\sqrt(n+4)} I chose my comparison series as \frac{1}{\sqrt(n)} and then ran a limit test...
  49. A

    What function should I use for comparison

    Homework Statement If i have the series ln(k)/k^2 would I compare it to 1/k^2? The reason why I'm confused is that ln(k)/k^2 > 1/k^2 so if its greater than a function that converges it doesn't tell us anything right? So I'm not sure what exactly to compare it with on account of the ln(k) in...
  50. F

    I am stupid and this book is being inconsistent with the Comparison test

    Homework Statement Before I state my problem, I need to quote this from my book http://img18.imageshack.us/img18/3748/comparisonnr.th.jpg Uploaded with ImageShack.us http://img806.imageshack.us/img806/8665/integraltest.th.jpg Uploaded with ImageShack.us See the problem here...
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