What is Ratio test: Definition and 101 Discussions

In mathematics, the ratio test is a test (or "criterion") for the convergence of a series





∑

n
=
1


∞



a

n


,


{\displaystyle \sum _{n=1}^{\infty }a_{n},}
where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alembert's ratio test or as the Cauchy ratio test.

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  1. T

    Using ratio test to test conditional convergence?

    Homework Statement So what I was taught was that if the lim of the ratio test is the series is always absolutely convergent. If it is >1 the series is always divergent. But if it is =1 then we don't know. So would that mean that all conditionally convergent series would have a limit = 1? I...
  2. P

    MHB Proving Convergence of e^z Series Using Ratio Test on Coefficients

    I am trying to prove e^z converges on all C. Here is my attempt. e^z=series(z^n/n!) use the ratio test on the coefficents 1/n! gives lim(1/(n+1))=0, which from rudin means radius of convergence R=1/0 -> R=infinity.
  3. I

    Question about one part of the Ratio Test proof

    Hi, there is a proof of the ratio test that I have seen a couple of times here: http://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspxI don't know how to use latex, so abs(x) will mean absolute value I am ok with this: abs( aN+1) < r abs(aN) I don't understand where the r2 come from in...
  4. D

    MHB Convergence of Ratio Test for Complex Numbers z in C

    $$\sum\limits_{n = 0}^{\infty}\frac{2\pi^{n+1}M}{(n+1)!}z^n$$ So we get $\lim\limits_{n\to\infty}\left|\frac{\pi z^n}{n+2}\right|$. This converges but I don't see how. z is in C.
  5. alexmahone

    MHB Convergence Condition for Applying Ratio Test to Power Series

    Given a power series, what is the condition on its coefficients that means the ratio test can be applied?
  6. B

    Ratio test for infinite series

    Homework Statement Investigate the convergence of the following series. (2n)!/(n!n!) The Attempt at a Solution Number one, I don't see how they get that if an = (2n)!, then an+1 = ((2n+2))!, it should be (2n+1)! Number two, I don't see how they go to the second step, why is the second step...
  7. alexmahone

    MHB Does the Converse of the Ratio Test Always Hold for Convergent Series?

    Is the converse of the ratio test true?
  8. M

    Using Ratio Test to Decide Convergent/Divergent

    Homework Statement The Attempt at a Solution I got that the limit goes to infinity using the Ratio Test and that would mean B. But I'm not 100% percent sure. Just looking for some to verify my answer if I'm correct or not.
  9. T

    Using ratio test to find radius of convergence

    Homework Statement Ʃ((x-3)^(n)) / (n*2^(n)) Homework Equations lim as n→ ∞ (An+1 / An) The Attempt at a Solution When dividing two fractions, invert the second and multiple to get what you see below. (x-3)^(n+1)/((n+1)*2^(n+1)) * (n*2^(n))/((x-3)^(n)) Do some cross...
  10. Rapier

    Using the Ratio Test Methodology to Determine Convergence

    Homework Statement Determine whether Ʃ(1→∞) n^2/e^n converges or diverges. Homework Equations L = lim (n→∞) abs [a_n+1/a_n] The Attempt at a Solution The prof was out of town so left us a "self-study" task. We're looking at the Ratio Test and I want to see if my methodology is...
  11. M

    A limsup inequality (showing that the root test is stronger than the ratio test)

    Homework Statement Show that if a_n > 0 for all n, \liminf{\frac{a_{n+1}}{a_n}} \leq \liminf{a_n^{1/n}} \leq \limsup{a_n^{1/n}} \leq \limsup{\frac{a_{n+1}}{a_n}}Homework Equations \liminf{a_n^{1/n}} \leq \limsup{a_n^{1/n}} \liminf{\frac{a_{n+1}}{a_n}} \leq \limsup{\frac{a_{n+1}}{a_n}} These...
  12. B

    Determining Series Convergence using the Ratio Test

    Homework Statement I'm asked to specifically use the Ratio Test (formula below) to determine whether this series converges or diverges (if it converges, the value to which it converges is not needed.) \sum_{n=1}^{\infty}\frac{n}{(e^n)^2} Homework Equations Ratio Test: If a_n is a sequence...
  13. L

    Convergence of series using ratio test

    Homework Statement assume summation of series An converges with all An>0. Prove summation of sqrt(An)/n converges Homework Equations The Attempt at a Solution I Tried using the ratio test which says if lim as n goes to infinity of |Bn+1/Bn|<1 then summation of Bn converges. I let Bn...
  14. S

    Ratio test inconclusive, what now?

    Homework Statement Determine divergence/convergence of this series: \sum\frac{(2n)!}{(n!)^{2}} * (\frac{1}{4})^{n} Homework Equations ratio test? The Attempt at a Solution I attempted to use the ratio test and the resulting limit was 1, which means the ratio test is inconclusive. So far, I...
  15. C

    Root test vs. ratio test question

    I am doing the following practice problem in prep for an exam: sum from n=0 to infinity: (3^n)/(n+1)^n The book says to use the ratio test on it, which I did, but would the root test also apply to this?
  16. M

    Proving convergence of factorial w/o Ratio Test

    Homework Statement Determine whether 1/n! diverges or converges. So far, we have only learned the comparison tests, p-series, geometric series, divergence test, and integral test, so I can only use these tests to prove it. Homework Equations N/a The Attempt at a Solution I...
  17. D

    Solving Ratio Test Limit with x and -3

    \begin{array}{l} \\ \mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{x^{n + 1}}\left( {\frac{1}{{{{\left( { - 3} \right)}^{n + 1}}}} - 1} \right)}}{{{x^n}\left( {\frac{1}{{{{\left( { - 3} \right)}^n}}} - 1} \right)}}} \right| = \mathop {\lim }\limits_{n \to \infty } \left| {\left(...
  18. D

    Ratio test for convergence of series 1/log(n)?

    Hi. I have been banging my head against this problem for a while and I just don't get it. Maybe (probably) it's something wrong with my logarithm-fu or limit-fu. I just registered to ask this because I couldn't find an answer anywhere, and I've been reading these forums for a while for other...
  19. M

    Ratio Test for series convergence factoring problems

    Homework Statement \Sigma2nn!/(n+2)! Homework Equations I'm using the ratio test because there are factorials but I'm a little stuck on whether or not to factor out The Attempt at a Solution lim 2n+1(n+1)!/(n+3)!*(n+2)!/2n(n)! After I set it up here I'm not sure of how to factor...
  20. N

    Mastering the Series Ratio Test: A Comprehensive Guide

  21. M

    Ratio Test inconclusive with factorial

    Homework Statement \sum(n-1)!/(n+2)! 2. The attempt at a solution I tried the ratio test and came up with the lim_{}n\rightarrow\infty n/(n+3) = 1 which gives no information on convergence or divergence. I'm trying to find absolute or conditional convergence so what else can I do?
  22. C

    Algebra behind the ratio test?

    Homework Statement \sum\limits_{i=0}^\infty \frac{i!}{5^i} Homework Equations The Attempt at a Solution (1) \sum\limits_{i=0}^\infty \frac{i!}{5^i} (2) = \lim_{i\rightarrow \infty}|{\frac{(i+1)!}{(5+1)^i} \cdot \frac{5^i}{i!}| (3) = \lim_{i\rightarrow...
  23. R

    D'Alembert Ratio Test: Convergence test

    Hi this is just a general question about using the ratio test for convergence. If I have to carry out the test to find out if something converges (and I don't need to find out if its absolutely converges, but just convergence), then can my answer to the test be negative? Or does the...
  24. Z

    Convergence or Divergence of Series with Square Roots

    Homework Statement Determine whether the series converges or diverges. \sum (\sqrt {k} - \sqrt {k - 1})^k Homework Equations The Attempt at a Solution (a_k)^\frac{1}{k} = \sqrt{k} - \sqrt{k - 1} What do I do here..? = \frac{1}{\sqrt{k} + \sqrt{k-1}} \to 0 ?
  25. M

    Ratio Test - Lim sup and lim inf version -

    Hello friends, Homework Statement lim sup |\frac{a_{n+1}}{a_{n}}| I know that this operation tests for convergence, but I don't understand how it's related to lim_{n -> \infty}|\frac{a_{n+1}}{a_{n}}| I understand that both will give the same result, but how is the latter different? lim...
  26. K

    Can we still use the ratio test if the sequence of ratios diverges to infinity?

    Let ∑ak be a series with positive terms. Ratio test: Suppose ak+1/ak -> c. If c<1, then ∑ak converges. If c>1, then ∑ak diverges. If c=1, the test is inconclusive. What if ak+1/ak diverges (i.e. ak+1/ak->∞)? Do we count this as falling into the case c>1? Can we say whether ∑ak converges...
  27. F

    Proof of the Ratio Test and the Triangle Inequality

    Homework Statement Prove: If the limit inf as k goes to infinity of abs(ak+1 / ak) > 1 then the sum from 1 to infinity of ak diverges Homework Equations The Attempt at a Solution So far I have this: Suppose lim inf abs(ak+1/ak) >1 then, there exists an r such that lim inf...
  28. F

    Proof of the Ratio Test and the Triangle Inequality

    Homework Statement Prove: If the limit inf as k goes to infinity of abs(ak+1 / ak) > 1 then the sum from 1 to infinity of ak diverges Homework Equations The Attempt at a Solution So far I have this: Suppose lim inf abs(ak+1/ak) >1 then, there exists an r such that lim inf...
  29. E

    Determining convergence of infinite series with factorial without ratio test

    Homework Statement Determine whether the series below is convergent or not: \sum 7*\frac{n!}{n^{n-10}} n=8 and the series goes to infinity (Sorry, I couldn't get the formatting correct.) Homework Equations n/aThe Attempt at a Solution Well, originally I thought the series was divergent...
  30. N

    Series Ratio Test: Answer 1/7 - Confirm?

    Hey there.. I've done the ration test for the series at the attachment.. i've got the answer 1/7 where the series is convergent.. can someone confirm my answer..?
  31. T

    Ratio Test for Radius of Convergence | Solving sum(5^n)((x-3)^n)/n"

    Homework Statement The problem is looking for the radius of convergence sum(5^n)((x-3)^n)/n n=1 Homework Equations The Attempt at a Solution
  32. J

    Did I Use the Ratio Test Correctly?

    Homework Statement Hi I want to see if you guys could check my work an tell me if i did it correct and also explain me how to cancel "look at the graph" Homework Equations The Attempt at a Solution
  33. J

    Cal 2 ratio test where is my mistake

    cal 2 ratio test "where is my mistake" Homework Statement I attached a pic I solved the problem but i get 4^1 and the book says 4^2 I don't know why? Homework Equations http://img690.imageshack.us/img690/1518/problem15.jpg The Attempt at a Solution
  34. F

    Proof of Ratio Test on Infinite Series

    Homework Statement I know that the ratio test can be proved using the geometric series knowledge, but I'm looking for a way to prove the ratio test without the geometric series at all. The reason is that I'm proving the geometric series convergence with the ratio test, and my professor...
  35. W

    Convergence Criteria for Series with Exponential and Polynomial Terms

    I have been stuck on this question for a while and have not got much success from class mates...can somebody please help? Using an appropriate convergence test, find the values of x \in R for which the following series is convergent: \sumn=1n \frac{1}{e^n * n^x} So, Un = \frac{1}{e^n...
  36. G

    Sequences ratio test, intro to real analysis

    Homework Statement Let X = (xn) be a sequence of positive real numbers such that lim(xn+1 / xn) = L > 1. Show that X is not a bounded sqeuence and hence is not convergent. Homework Equations Definition of convergence states that for every epsilon > 0 there exist some natural...
  37. D

    Does the Ratio Test Guarantee Divergence? Proving with Bernoulli's Inequality

    Homework Statement Show that if Lim|\frac{a_{n+1}}{a_{n}}| = L > 1, then {a_{n}\rightarrow \infty as n\rightarrow\infty Also, from that, deduce that a_{n} does not approach 0 as n \rightarrow \infty . Homework Equations The book suggests showing some number r>1 such that for some...
  38. H

    .Determining Convergence/Divergence of 100n/2^n with Ratio Test

    Homework Statement Use the ratio test to determine convergence or divergence of 100n/2^n Homework Equations p= lim┬(n→∞)[a_(n+1)/a_n] p<1 ,convergent p>1 ,divergent p=1 ,ratio test fails The Attempt at a Solution I'm not even sure if I'm doing this right, but i get...
  39. H

    Likelyhood ratio test hypotheses and normal distribution

    Homework Statement Given the normal distribution X_{ij} \sim N(\mu_i, \omega^2) where i = 1,2 and j = 1,...,n deduce that H_{0\mu}: \mu_1 = \mu _2 The Attempt at a Solution Do I take in the Likelyhood function here? and use it to analyse the case? Sincerely Hummingbird p.s. I have...
  40. R

    Solve Ratio Test Problem: Factorial Cancelling

    Homework Statement heres the problem: http://img181.imageshack.us/img181/581/59319587uw4.png Homework Equations The Attempt at a Solution where I'm at is in the pic, i have the problem all ready to cancel, but I'm really confused about factorial cancelling. Just need what...
  41. R

    What is the convergence of a power series using the ratio test?

    Homework Statement I've tried to apply the ratio test to a problem that is a power series. here's the problem as a pic: http://img152.imageshack.us/img152/2751/35685690oj3.png Homework Equations The Attempt at a Solution I've gotten so far as you can see in the pic, I've...
  42. Q

    Don't understand how they simplified this (ratio test)

    Don't understand how they simplified this...(ratio test) I'm studying for my exam and I was looking at this example: I'm not really sure how they get from here : to here: If someone could explain how they simplified this to me that would be fantastic...i'm really trying to understand...
  43. F

    Is it ok to do this with the ratio test for series?/

    Is it ok to do this with the ratio test for series??/ I had the series from 1 to infinity of: n(-3)^(n)/(2^(n-1)) by applying the root test, i got: lim as n-->infinity [ 3(n+1)/2n] , so put the 3/2 outside and let the (n+1)/n be n/n --> which means the limit would yield 3/2...
  44. F

    How simplifying this ratio test for series?

    [SOLVED] How simplifying this ratio test for series? Basically, we are asked if (n!n^2)/(2n)! converges absolutely... I got to the point where lim as n --> infinity of [(n+1)!(n+1)^2]/(2(n+1))! X (2n)!/n!n^2 by the ratio test. But I don't know how to manipulate the factorials...
  45. P

    How do you prove the validity of the ratio test?

    It dosen't seem intuitive that if the ratio of the last two consecutive terms is less than 1 then it is convergent and viceversa if divergent when adding an infinite number of terms.
  46. S

    Using root test and ratio test for divergence

    Homework Statement Does this series converge or diverge? Series from n=1 to infinity n(-3)^(n+1) / 4^(n-1) Homework Equations The Attempt at a Solution Okay, I've tried it both ways. Ratio test: lim n --> inf. ((n+1)*(-3)^(n+1)/4^n) / (n * (-3)^n / 4^(n-1)) Now...
  47. B

    Radius and Interval of Convergence for (3^n x^n)/(n+1)^2 Series

    Have to find the radius of convergence and interval of convergence, the series is (3^n x^n ) / (n+1)^2, did the ratio test and found the radius of convergence to be the 1/3. now for finding the interval of convergence I plug in -1/3 and 1/3 into x and find out if it converges or not For...
  48. M

    Does the Ratio Test guarantee convergence for this infinite series?

    Hi all! Here's something I'm having difficulty seeing: Suppose u_n > 0 and \frac{u_{n+1}}{u_n} \leq 1-\frac{2}{n} + \frac{1}{n^2} if n \geq 2 Show that \sum{u_n} is convergent. I'm not sure how to apply the ratio test to this. It looks like I would just take the limit. I get: lim_{n...
  49. B

    Generalized likelihood ratio test

    This is the problem (t for theta): X ~ Expo(t) = t * e ^ (-t * x), x>0, t >0 0 otherwise Test H0: t <= 1 vs. H1: t >1 using the generalized likelihood ratio test where you have a random sample from X {X1, X2, ... , X50} and the sum of all Xi = 35. Use alpha =...
  50. C

    Ratio test, why does the (-1)^(n+1) disappear?

    In the book while doing the ratio test: as n --> infinity why do they just have (-1)^(n+1) just disappear in the next step, since it oscillates between -1 and 1, I don't understand how u could just make it disappear in computations. Isn't the limit as n -> infinity , equal to D.N.E.?
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