What is Ratio test: Definition and 101 Discussions

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





{\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. Ackbach

    MHB Binomial Distribution: Likelihood Ratio Test for Equality of Several Proportions

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: A survey of voter sentiment was conducted in four midcity political wards to compare the fraction of voters favoring candidate $A.$ Random samples of $200$ voters were polled in each of the...
  2. Ackbach

    MHB Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of independent random samples of sizes $n$ and $m$ selected from normal distributions with means $\mu_1$ and $\mu_2$ and common...
  3. W

    Understanding the Ratio Test for Series and Its Applications

    So I am having some difficulty expressing this series explicitly. I just tried finding some terms ##b_{0} = 5## I am assuming I am allowed to use that for ##b_{1}## for the series, even if the series begins at ##n=1##? With that assumption, I have ##b_{1} = -\frac {5}{4}## ##b_{2} = -...
  4. M

    MHB Proving Boundedness of Seq. $(a_n)$ Given Bound Seq. $(\frac{a_{n+1}}{a_n})$

    Hey! :giggle: Show for each sequence $(a_n)\subset (0, \infty)$ for which the sequence $\left (\frac{a_{n+1}}{a_n}\right )$ is bounded, that $\sqrt[n]{a_n}$ is also bounded and that $$\lim \sup \sqrt[n]{a_n}\leq \lim \sup \frac{a_{n+1}}{a_n}$$ I have done teh following: The sequence $\left...
  5. C

    I Akaike Information Criterion Vs Likelihood Ratio Test

    Hello, I want to understand the difference between both goodness-of-fit tests, I would be glad if you could help me: Akaike Information criterion is defined as: ## AIC_i = - 2log( L_i ) + 2K_i ## Where ##L_i## is the likelihood function defined for distribution model ##i## . ##K_i## is the...
  6. D

    Checking the convergence of this numerical series using the ratio test

    ## \sum_{n=0}^\infty \frac {(2n)!}{(n!)^2} ## ##\lim_{n \rightarrow +\infty} {\frac {a_{n+1}} {a_n}}## that becomes ##\lim_{n \rightarrow +\infty} {\frac { \frac {(2(n+1))!}{((n+1)!)^2}} { \frac {(2n)!}{(n!)^2}}}## ##\lim_{n \rightarrow +\infty} \frac {(2(n+1))!(n!)^2}{((n+1)!)^2(2n)!}##...
  7. F

    Infinite Series (The Ratio Test)

    I found that ρn = √(2n+1)/(n+1). Then, I found ρ = lim when n→∞ |(1/n) (√(2n+1))/((1/n) (n+1))| = 0 Based on this result I concluded the series converges; however, the book answer says it diverges. What am I doing wrong?
  8. karush

    MHB Ratio Test Determines Divergence: 11.6.1

    Determine Convergence or divergence and test used $\displaystyle\sum_{n=1}^{\infty} \dfrac{1+4^n}{1+3^n}$ W|A says diverges using ratio test so $\therefore L=\lim_{n \to \infty}\left|\dfrac{a_n+1}{a_n}\right|>1$ Steps $\displaystyle L=\lim_{n \to \infty}\left|...
  9. karush

    MHB 11.6.8 determine convergent or divergence by Ratio Test

    Use the Ratio Test to determine whether the series is convergent or divergent $$\sum_{n=1}^{\infty}\dfrac{(-2)^n}{n^2}$$ If $\displaystyle\lim_{n \to \infty} \left|\dfrac{a_{n+1}}{a_n}\right|=L>1 \textit{ or } \left|\dfrac{a_{n+1}}{a_n}\right|=\infty...
  10. MathematicalPhysicist

    I A question regarding the ratio test for limits

    So we have the theorem: if ##a_n>0## and ##\lim_{n\to \infty} a_{n+1}/a_n = L## then ##\lim_{n\to \infty} a_n^{1/n}=L##. Now, the proof that I had seen for ##L\ne0## that we choose ##\epsilon<L##. But what about the case of ##\epsilon>L##, in which case we have: ##a_{n+1}>(L-\epsilon)a_n## but...
  11. S

    Does the series 4/k(lnk)^2 converge or diverge?

    Homework Statement ##\sum_{k=0}^\infty \frac 4 k(\ln k)^2 ## Homework EquationsThe Attempt at a Solution I tried to solve it using the integral test but since it's not continuous it doesn't work.
  12. saybrook1

    Don't understand this limit change in a ratio test

    Homework Statement I would like to understand how the limit was changed in the ratio test from step 1 to step 2 in the image that I've posted. I thought that the denominator would look like (2/n+2)(2/n+1) in step 2 since it looks like we are just turning the n's into reciprocals. Any help here...
  13. T

    A Improving intuition on applying the likelihood ratio test

    I am trying to better understand likelihood ratio test and have found a few helpful resources that explicitly solve problems, but was just curious if you have any more to recommend. Links that perhaps work out full problems and also nicely explain the theory. Similar links you have found...
  14. B

    Radius of Convergence for Ratio Test in Calculus Questions

    Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...
  15. B

    Absolutely Convergent, Conditionally Convergent, or Divergent?

    Homework Statement ∞ Σ (-1)n-1 n/n2 +4 n=1 Homework Equations lim |an+1/an| = L n→∞ bn+1≤bn lim bn = 0 n→∞ The Attempt at a Solution So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo. I found that...
  16. karush

    MHB Ratio Test for Sum $\tiny{}$

    $\tiny{}$ \begin{align*} \displaystyle S_{84}&=\sum_{k=1}^{\infty} \frac{(4x)^k}{5k}\\ \end{align*} $\textsf{ ratio test}$ $$\frac{a_{n+1}}{a_n} =\frac{ \frac{(4x)^{k+1}}{5(k+1)}}{ \frac{(4x)^k}{5k}} =\frac{4xk}{k+1} $$ $\textsf{W|A says this converges at $4|x|<1 $ so how??}$
  17. karush

    MHB Supposed to use the root and ratio test

    $\tiny{206.b.46}$ \begin{align*} \displaystyle S_{46}&=\sum_{k=1}^{\infty} \frac{2^k}{e^{k}-1 }\approx3.32569\\ % e^7 &=1+7+\frac{7^2}{2!} %+\frac{7^3}{3!}+\frac{7^4}{4!}+\cdots \\ %e^7 &=1+7+\frac{49}{2}+\frac{343}{6}+\frac{2401}{24}+\cdots \end{align*} $\textsf{root test}$...
  18. T

    MHB Ratio test with an integer power of an in numerator

    I have $$\sum_{n = 1}^{\infty} \frac{2^n}{n^{100}}$$ and I need to find whether it converges or diverges. I can use the ratio test to get: $$\lim_{{n}\to{\infty}} \frac{2^{n + 1}\cdot n^{100}}{2^n \cdot (n + 1)^{100}}$$ But I'm not sure how to get the limit from this. I know the limit of...
  19. mertcan

    I Power series ratio test

    hi, If you look at my attachment you can see that the book express that for the situation of x=+,-(1/L) we need further investigation. It means being converged or diverged is not precise. I would like to ask: Is there remarkable proof that if x=+,-(1/L) convergence or divergence is not...
  20. erbilsilik

    Bose function convergence

    Homework Statement How can I show that this series is convergent for z=1 and z<1 and divergent for z>1 $$\sum _{p=1}^{\infty }\dfrac {z^{p}} {p^{3/2}}$$ Homework Equations http://tutorial.math.lamar.edu/Classes/CalcII/RatioTest.aspx The Attempt at a Solution Using the ratio test I've...
  21. J

    Finding the Radius of Convergence through Ratio Test

    Homework Statement Let f(x)= (1+x)4/3 - In this question we are studying the Taylor series for f(x) about x=2. This assignment begins by having us find the first 6 terms in this Taylor series. For time, I will omit them; however, let's note that as we continuously take the derivative of this...
  22. F

    Ratio Test for Finding the 1(circled part) - Understanding the Process

    Homework Statement where does the 1(circled part) come from ? is it a mistake ? Homework EquationsThe Attempt at a Solution
  23. P

    MHB Yonglie's question via email about ratio test

    First, because the series is positive term, we don't have to worry about absolute values. Now $\displaystyle \begin{align*} a_n = \frac{2n + 3}{4n^3 + n} \end{align*}$ and $\displaystyle \begin{align*} a_{n + 1} &= \frac{2\left( n + 1 \right) + 3}{4 \left( n + 1 \right) ^3 + n + 1} \\ &=...
  24. R

    Finding the radius of convergence of a power series

    Homework Statement Σ(n=0 to ∞) ((20)(-1)^n(x^(3n))/8^(n+1) Homework Equations Ratio test for Power Series: ρ=lim(n->∞) a_(n+1)/a_n The Attempt at a Solution I tried the ratio test for Power Series and it went like this: ρ=lim(n->∞) (|x|^(3n+1)*8^(n+1))/(|x|^(3n)*8^(n+2)) =20|x|/8 lim(n->∞)...
  25. shanepitts

    Applying the ratio test

    Below is a screen shot of a solution to a problem. The part I don't fathom is after the ratio test is applied to the denominator. How can, noting that an+1, (2n-1) become (2n-1)(2n+1) and not just (2(n+1)-1)=2n+1? Thank you in advance
  26. C

    Quick question about Ratio Test for Series Convergence

    Homework Statement [/B] This is the question I have (from a worksheet that is a practice for a quiz). Its a conceptual question (I guess). I understand how to solve ratio test problems. "Is this test only sufficient, or is it an exact criterion for convergence?" Homework Equations Recall the...
  27. M

    Learning from Mistakes: Ratio Test Problem Solving

    Homework Statement I'm reviewing for a test and working on the practice problems for the ratio test that Pauls Online Notes gives. So here is given problem: Here is his solution for the problem: 2. The attempt at a solution I worked this out before I looked at the solution and I got it wrong...
  28. A

    I can't seem to understand the ratio test proof

    Hi everyone, I'm currently taking Calc II course and I'm kind of stuck in this ratio test proof thing. Homework Statement http://blogs.ubc.ca/infiniteseriesmodule/appendices/proof-of-the-ratio-test/proof-of-the-ratio-test/ I'm trying to understand the proof, but there are some parts that I...
  29. C

    Ratio Test Radius of Convergence

    Homework Statement ∑ x2n / n! The limits of the sum go from n = 0 to n = infinity Homework EquationsThe Attempt at a Solution So I take the limit as n approaches infinity of aa+1 / an. So that gives me: ((x2n+2) * (n!)) / ((x2n) * (n + 1)!) Canceling everything out gives me x2 / (n + 1)...
  30. A

    MHB Ratio test and root test

    Hello. How do I determine whether to use ratio test or root test in determining whether a series is convergent or divergant? For example, in this problem, ratio is used for no.1 and root test for no.2. Why is that? I need explanation, please.
  31. M

    Ratio test and root test

    Hello. How do I determine whether to use ratio test or root test in determining whether a series is convergent or divergant? For example, in this problem, ratio is used for no.1 and root test for no.2. Why is that? I need explanation, please.
  32. M

    Proving part of the ratio test

    This is not a homework problem. I'm doing it for fun. But it is the kind that might appear on homework. Homework Statement I'm trying to prove that if lim n→∞ |an+1/an| = L < 1, then \Sigma an converges absolutely and therefore converges. Homework Equations The Attempt at a Solution Here's...
  33. D

    Ratio Test Problem: Homework Statement & Solutions

    Homework Statement See attached image. (it should say "ratio" not "ration") Homework Equations Ratio series test: An+1/An The Attempt at a Solution I have worked this problem over and over and continue to get the same solution. Some guy worked it on the board a couple of days ago and got...
  34. H

    Ratio test for finding radius of convergence

    Homework Statement I've found that the typical way for using ratio test is to find the limit of an+1/an However, my tutor said that radius of convergence can be found by finding the limit of an/an+1 and the x term is excluded. For example:Finding the interval of convergence of n!xn/nn my...
  35. M

    Simplifying a series with the ratio test

    Homework Statement Determine if the following series is divergent or convergent: ## ∑_1^∞ \frac {(2)(4)(6)...(2n)}{n!} ## 2. The attempt at a solution I understand this can be simplified to: ## ∑_1^∞ \frac {(2^n)(n!)}{n!} ## This can easily be seen to be divergent. But when I...
  36. Fernando Revilla

    MHB Root or Ratio Test: Interval of Convergence

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  37. J

    Limit Ratio Test: Solving $$\sum_{n=1}^{\infty}\frac{1}{2^n}$$

    Homework Statement $$ \sum_{n=1} ^\infty\frac{1} {2^n} $$ Homework Equations The Attempt at a Solution I know just by looking at it that it converges no problem. You do the ratio test and you get something of the form \displaystyle\lim_{n\rightarrow \infty}...
  38. Fernando Revilla

    MHB  Use the Ratio Test for Convergence/Divergence

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  39. D

    Ratio test for math convergence

    Homework Statement show ## \sum \frac{x^{2}}{(1+x^{2})^{n}} ## converges uniformly on R Homework Equations The Attempt at a Solution I know by ratio test it is absolutely convergent for all x in R. I am guessing you use m-test. However I do not really understand how...
  40. F

    Ratio test (convergent or divergent?)

    Homework Statement ∞ Ʃ n / 2^n n=1 Homework Equations ratio test lim |a(n+1) / a(n)| n->∞ The Attempt at a Solution I have the answer and the steps its just there's one part I am confused on, first I just apply n+1 to all my n terms, which gives me, ∞ Ʃ...
  41. C

    MHB Ratio Test Questions/ Series Convergence

    I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3) and I am doing the Ratio Test. I found the limit approaches 1 but is less than 1. Does this mean that the limit = 1 or is < 1? I am somewhat confused since this changes it from inconclusive to convergent.
  42. A

    Using the ratio test to figure if the series is convergent

    Homework Statement \sum ftom n=1 to \infty (-2)n/nn. The Attempt at a Solution limn->\infty | (-2)n+1/(n+1)n+1) x nn/(-2)n | = |-2|limn->\infty |(n/n+1)n*(1/n+1) | If it were only (n/n+1) then would the answer be 2e? Either way, how do you sole this the way it is?
  43. L

    Using the Ratio Test to see if a series converges or diverges?

    Homework Statement Use the Ratio Test for series to determine whether each of the following series converge or diverge. Make Reasoning Clear. (a) \sum^{∞}_{n=1}\frac{3^{n}}{n^{n}} (b) \sum^{∞}_{n=1}\frac{n!}{n^{\frac{n}{2}}} Homework Equations...
  44. P

    Sine Ratio Test: Show Convergence w/o L'Hopital

    Hi, Without using l'hopital, how may I show that sin[(10pi)/(n+1)^2] / sin[(10pi)/n^2] converges?
  45. B

    Solve Ratio Test Problem: ∞ Ʃ (n!)^3/3(n)!

    ∞ Ʃ (((n)!)^3)/(3(n))! Use the ratio test to solve n=1 So first i put it into form of (n!)^3/3n!, then applied ratio test. from ratio got ((n+1)!)^3/(3n+1)! times (3n)!/(n!)^3 from here I am on shaky ground i go reduce the terms to (n!)^3(n+1)^3/(3n!)(3n+1) times...
  46. B

    Proof of the ratio test

    I am trying to understand something in the proof of the ratio test for series convergence. If a_{n} is a sequence of positive numbers, and that the ratio test shows that \lim_{n→∞}\frac{a_{n+1}}{a_{n}} = r < 1, then the series converges. Apparently, the proof defines a number R : r<R<1...
  47. M

    Use ratio test to find radius and interval of convergence of power series

    Homework Statement Use the ratio test to find the radius of convergence and the interval of convergence of the power series: [[Shown in attachment]] Homework Equations an+1/an=k Radius of convergence = 1/k Interval of convergence: | x-a |∠ R The Attempt at a Solution I...
  48. B

    Ratio Test for Series Homework: Author's Solution & Attempt at a Solution

    Homework Statement I attached a file that includes the author's solution, and some of my work. Homework Equations The Attempt at a Solution
  49. A

    Integral test and ratio test on haromonic series.

    So harmonic series diverges because of the integral test but if I try it on ratio test = (1 / ( x+1 )) / (1 / x) = x / (x + 1) and this is less than 1 so shouldn't it converge?
  50. A

    Ratio Test, SUPER , help?

    Ratio Test, SUPER URGENT, help? Consider the series ∑ n=1 to infinity of asubn, where asubn = [8^(n+4)] / [(8n^2 +7)(5^n)] use the ratio test to decide whether the series converges. state what the limit is. From the ratio test I got the limit n-> infinity of [8^(n+1+4)] / [(8(n+1)^2...