1/4/2024 0 Comments Cauchy sequence divergenceThose are convergent, andī) for my Sn/S'n, I’d like to have the same dominant term in the numerator and the denominator. For example, a p-series with n>1, or a geometric series with |r|1).įor example: S = Σ(n=1 to infinity)((n+1)/(n³+6n+3)).ġ) The dominant term of the denominator (n³) is larger than the dominant term of the numerator (n), so my intuition says that S should converge.Ģ) a) Since I am dealing with rational functions (ratios of polynomials), a p-series with n>1 would make a good S'. However, I do have a recipe to approach the proof of a series’ convergence/divergence with the limit comparison test.ġ) I start with an intuition: does S looks like is converges or diverges? Let’s say it looks like it converges.Ģ) Then I look for another series S' such that:Ī) I know that S' converges. Oresme’s proof is very witty and I believe finding such proofs requires patience, serendipity, creativity and mathematical experience. I suggest you do a search on the convergence of 1/n² to get more background so you can feel confident in your own mind why this is so. There are several proofs, one using the integral test, another uses the Cauchy Condensation test. In a similar manner, the terms add so little to the sum that they do not impact the total. Do you see how we can keep doing this process infinitely many times and NEVER get to b. Now to get to b you need to go half way from a' to b. To get to point b you first need to go half the distance from a to b. That is, each term in 1/n² decreases at a sufficiently fast rate that even though you continue to add to the sum, what you are adding is so infinitesimally small that it makes no difference to the final sum. The reason that 1/n² converges has to do with how fast each term is getting smaller. Was just a very simple simulation, here is the code in R:įirst, the series 1/n and 1/² both have the same number of terms, an infinite number. This seems to indicate the series is diverging (if it were lowering I'd say it would eventually converge). This increase by 2.3 per order of magnitude does change very minimally in a positive as opposed to a negative direction. The increase here has gone up to 2.3.Īs the order of magnitude increase approaches very large numbers (I ran a simulation going to 9 orders of magnitude, a steady increase of 2.3 is noticed. However, if we take 100,000 terms, we end up with approximately 12.09, compare this to one order of magnitude less, which is only 10,000 terms and we end up with approximately 9.79. The difference between these two numbers is about 2.26. If we take 100 terms, we end up with 5.19. Interestingly I noticed that for each increase in order of magnitude of the number of terms, the sum of the series increases by approximately 2.3, however this number seems to increase rather than decrease, indicating a non-convergence.įor example, say we take 10 terms, we end up with approximately 2.92. Hank, your observation spurred me to find an answer myself, so I ran some simulations.
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