# 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.

View More On Wikipedia.org
1. ### 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. ### 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. ### 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} = -...

24. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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...