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.
$\newcommand{\szdp}[1]{\!\left(#1\right)}
\newcommand{\szdb}[1]{\!\left[#1\right]}$
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...
$\newcommand{\szdp}[1]{\!\left(#1\right)}
\newcommand{\szdb}[1]{\!\left[#1\right]}$
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...
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} = -...
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...
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...
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?
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|...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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} \\ &=...
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->∞)...
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
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...
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...
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...
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)...
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.
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.
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...
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...
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...
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...
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}...
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...
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...
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.
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?
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...
∞
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...
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...
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...
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?
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...