Comparison test Definition and 106 Threads

Homework Statement A few examples done in class that I didn't understand are as follows: (Find out if convergent or divergent) integral 0-->infinity ( x / (x^3 + 1) ) integral 1-->infinity ( (2+e^(-x) ) / x ) Homework Equations None The Attempt at a Solution Basically, I have NO idea...

Homework Statement Using the limit comparison test to determine convergence or divergence (1) \sum^{\infty}_{n=1} \frac{(ln(n))^{3}}{n^{3}} (2) \sum^{\infty}_{n=1} \frac{1}{\sqrt{n} ln(x)} (3)\sum^{\infty}_{n=1} \frac{(ln(n))^{2}}{n^{\frac{3}{2}}} Homework Equations...

Homework Statement Determine if the following integrals are convergent or divergent. Explain why. \int^{1}_{0} \frac{1}{1-x^{4}} dx The Attempt at a Solution I've tried using Comparison Test, using f(x) = \frac{1}{1-x^{4}} and\; g(x) = \frac{1}{1-x}, 0 \leq f(x)\leq g(x) in ] 0,1 [ and I...

The Problem: Let a(n) = (n^2)/(2^n) Prove that if n>=3, then: (a(n+1))/(a(n)) <= 8/9 By using this inequality for n = 3,4,5,..., prove that: a(n+3) <= ((8/9)^n)(a(3)) Using the comparison test and results concerning the convergence of the geometric series, show that: The...

I was just wondering for the limit comparison test does it matter which function is on the top and bottom?

Note: This is not strictly a homework problem. I'm just doing these problems for review (college is out for the semester) - but I wasn't sure if putting them on the main part of the forum would be appropriate since they are clearly lower-level problems.(Newbie) Homework Statement The...

Series Comparison Test, URGENT help? Each of the following statements is an attempt to show that a given series is convergent or divergent using the Comparison Test (NOT the Limit Comparison Test.) For each statement, enter C (for "correct") if the argument is valid, or enter I (for...

I'm given the following: 3/(n(2^(n-1))) I have to determine convergence using the limit comparison test. I've proved its convergent using the ratio test but am struggling with which term do I divide the above for the limit comparison test. Help?

Homework Statement \sum^{∞}_{1}1/n^{n} Homework Equations Direct comparison test The Attempt at a Solution Since the main factor in the equation is the exponent that would be changing as n goes to infinity, I know that from the p series as p > 1 the the series converges. So I know...

Homework Statement For the sum from n=1 to ∞ (1/(sqrt(n^2+1)), I know you can use the limit comparison test to show that it is divergent but I was wondering if it is possible to compare this with 1/(2n)? I am not sure if 1/(2n) is always less than (1/(sqrt(n^2+1)) within those bounds. How...

Homework Statement The original question is posted on my online-assignment. It asks the following: Determine whether the following series converges or diverges: \sum^{\infty}_{n=1}\frac{3^{n}}{3+7^{n}} There are 3 entry fields for this question. One right next to the series above...

Homework Statement a) Test the following series for convergence using the comparison test : sin(1/n) Explain your conclusion. Homework Equations The Attempt at a Solution i must show f(x)<g(x) in order for it to converge other wise divergence. g(x) = 1/n sin(1/n) >...

Use a comparison test to determine whether the series \sum (n+1)/(n^{2}+n+1) diverges or converges. I started out by simplifying the series to 1/n+1 and then from there I compared it to 1/n, which converges. 1/n is greater than 1/n+1 so based on the comparison test, the original series...

Homework Statement The question in the book is: "Use the sum of the first 10 terms to approximate the sum of the series. Estimate the error. My problem is estimating the error I'm looking for. I just need help with finding the integral. Homework Equations ((sin n)^2)/(n^3) The...

Homework Statement Does the following interval diverge? \int^9_1 \frac{-4}{\sqrt[3]{x-9}} Homework Equations The Attempt at a Solution Well.. I've used the following function that I think is always less than the above function to prove that the function above DOES NOT diverge (by...

Homework Statement Determine whether the series converges or diverges. What I would like is some type of information on how to continue the problem. Homework Equations ∞ Ʃ √(n+1)/2n^2+n+1 n=1 The Attempt at a Solution I was thinking of doing a comparison test by doing...

\sum_{n=1}^\infty \frac{1}{n!} I understand what n! means, but I have no clue what to compare this to. It is obvious to me that the sum converges, but I'm not sure how to prove it. I assume I would compare it to a p-series but I need help. Thanks!

integral 1/(1+x^2)dx from 0 to infinity I decided to compare that to 1/(1+x) (saying 1/(1+x) > (1/(1+x^2)) but this diverges when the original equation converges. Can someone explain why the integral 1/(1+x) was not a proper choice and what the process would be to find a correct comparison.

2 problems, I need to use the direct comparison test with either a p series or a geometric series 1)series of j^2/(j^3 +4j +3) I thought of comparing it to j^2/J^3 which comes out to 1/j, but that dosent work, my teachers answer is you compare it to 1/5j 2) series of sqrt(q)/(q+2) I would...

Homework Statement Determine the Convergence or Divergence of \sum_{n=1}^\infty\left(sin(1/n)\right) Homework Equations limit comparison test The Attempt at a Solution I don't know what to compare this series to, I tried the harmonic series to get sin(1/n)/(1/n) = nsin(1/n) which...

Homework Statement Summation from n=1 to infinity: (e^(1/n)) / n Homework Equations The Attempt at a Solution Can someone point out what criteria I should be considering when trying to determine which test to use? I was looking at a comparison test as a way to go on this one...

Problem: Summation from n=1 to infinity: 1/(n^3 + n^2) I understand, for example, another problem wherein it is 3/(4^n + 5) what you would compare that one to but how do you go about breaking this one up with two "n" terms? Are you supposed to, in general, pick the largest value of n? So...

Homework Statement Before I state my problem, I need to quote this from my book http://img18.imageshack.us/img18/3748/comparisonnr.th.jpg Uploaded with ImageShack.us http://img806.imageshack.us/img806/8665/integraltest.th.jpg Uploaded with ImageShack.us See the problem here...

Homework Statement determine whether the series diverges or converges (∞,n=1) ∑ (n2 - 1)/(3n4 +1) Homework Equations The Attempt at a Solution (∞,n=1) ∑ (n2 - 1)/(3n4 +1) an = (n2 - 1)/(3n4 +1) i thought bn should be n2/3n4 = 1/3n2 lim n-->∞ an/bn = (n2 - 1)/(3n4 +1) * 3n2 lim n-->∞...

Homework Statement Determine whether or not the integral from 0 to 1 of (5ln(x)) / ( x^(3/2) ) converges or not. Homework Equations The Attempt at a Solution I just need to know which end of the integral they are talking about. As x=>0, y=>-infinity. As x=>1, y=>1. I'm assuming...

Homework Statement Determine whether or not the improper integral from 0 to infinite of (e^x)/[(e^2x)+4] converges and if it does, find it's definite value. Homework Equations The Attempt at a Solution I missed the lecture on the Comparison Test, so I'm essentially useless. I...

Homework Statement The summation of n/lnn from n=1 to infinity Homework Equations Ratio and Comparison test The Attempt at a Solution http://i55.tinypic.com/2jxuue.jpg Alright so the way labeled "right way" on the picture is the right way according to the book but isn't the...

Just a general question, but I find it hard to come up with a b[n] to compare to a[n]. When the book does it, they come up with stuff to compare to a[n] that I would have never thought of. Is there any criteria, things to look for, etc., for coming up with a b[n] to compare to a[n]?

Homework Statement \sum_{n=1}^\infty \frac{1+4^n}{1+3^n} where a_n= \frac{1+4^n}{1+3^n} and b_n= \frac{4^n}{3^n} Homework Equations I know how to do this problem, you take the limit as "n" goes to infinity of a_n/b_n ... which after a good amount of algebra ends up being 1...

Finding "a" and "b" in an infinite series limit comparison test Homework Statement \sum_{n=1}^\infty \frac{\sqrt{n+2}}{2n^2+n+1} How do I identify my a_n and my b_n? In this particular problem you need to use the Limit comparison test which is your "a_n" divided by your "b_n". I...

Homework Statement show whether \sum (n+5)/(n3-2n+3) is convergant of divergant Homework Equations limit comparison test, lim an/bn = c where c > 0 The Attempt at a Solution an is given let bn = n/n3 so then: lim (n+5)/(n3-2n+3) n/n3 = lim (n+5)n3 /...

Homework Statement Explain why the Direct Comparison Test allows us to use the inequality Ln n < n^(1/10) even though it is not true for a great many n values. Homework Equations The Attempt at a Solution I looked at the graphs of Ln (n) vs. n^(k)

Homework Statement Finally done with stupid improper integrals (calc 2 over summer is hard work, moving super fast) and now I'm sequences and series and what have you. I have another comparison test problem Instructions are to find out if sequence converges or diverges, and find limit if it...

Homework Statement I'm just curious as to how to think about the following form of equation. Homework Equations \int_{3}^{\infty } \frac{1}{x + e^x} \,dx The Attempt at a Solution What you're trying to do is to test it; \frac{1}{x \ + \ e^x} \ < \ \frac{1}{x} \frac{1}{x}...

Can the limit comparison test work on alternating series? I looked at the conditions online and it said that the series a_n and b_n must be > 0 in order to use it. To be specific it said, "Suppose that a_n > 0 and b_n>0 for all n> or = to N (N a positive integer)." I asked my teacher about this...

Homework Statement Find if the sum from n = 1 to infinity of (n^n)/n! diverges or not. Homework Equations p = an+1/an The Attempt at a Solution Using the comparison test I get to the point where p_n = (n+1)^(n+1) / [(n+1) n^n] Shouldnt p just be 0, don't (n+1)^(n+1) and n^n...

Homework Statement I need to show that: \sum_{n=1}^\infty \frac{n^{2}}{2^n} converges. I know I can compare it with the larger convergent geometric series: \sum_{n=1}^\infty \frac{1.5^{n}}{2^n} Which is larger for all terms for n> 13. My question is, I found this "13" through trial and...

Homework Statement \sum^{\infty}_{n=1} \frac{e^{n}+n}{e^{2n}-n^{2}} Homework Equations I have to use either the Comparison Test or the Limit Comparison Test to show whether the series converges or diverges. The Attempt at a Solution a_{n} = \frac{e^{n}+n}{e^{2n}-n^{2}} b_{n} =...

Homework Statement Using the comparison test determine if the infinite series for sin(3/n^2) converges or diverges. The Attempt at a Solution Well... these are pretty straight forward, and it's pretty obvious that this is convergent, but I'm having trouble applying the...

I read a proof for showing that the harmonic series is a diverging one. This particular one used a comparison test: 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + ... + 1/16 + ... 1/2 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + 1/16 +... + 1/16 + ... Each term in the second series is <...

sum(1/n!,n,1,inf) the only thing i can think of is 1/n^2>=1/n! , 1/n^2 being a convergent p series since p >1 thus /n! is convergent

14. sum(n^(1/2)/(n^2+1),n,1,inf) b=n^(1/2)/(n^2+2)<=n^(1/2)/(n^2+1)=a b is conv since lim as n-> inf b = 0 since n^(1/2)<n^2+2 for x>=1 thus a is conv i have a feeling this is a shakey way to do this if its even correct somone pleae clarify the solution to this problem

Homework Statement I have two relatively similar problems: 1.) Sigma n=1 to infinity ((ln n)^3 / n^2) 2.) Sigma n=1 to infinity (1 / sqrt(n) * (ln n)^4) I'm to prove their convergence or divergence using either the direct comparison test or the limit comparison test. I understand...

http://img205.imageshack.us/img205/5117/summation.jpg I think next step is: http://img205.imageshack.us/img205/7044/summation2.jpg but the questions are: how & why? why is that the next step? how do I solve such problems?\how do I choose the second part?

hey guys I'm kind of stuck on this idea of the comparison test. i understand the process and how it is done but i don't seem to understand how to find the second series(typically called bn) to compare with the original. get what i mean? thanks! =)

Does anyone have any knowledge on these? I look at an example of a direct or limit comparison and i see what they are doing, but I have no idea how they got the "comprable" term... for example the sum (n = 1) to infinity n/(n^2 +1) you can use either test, but how do you choose what to...

I am confused about the right formula for this. Is it R_{n} \leq T_{n} \leq \int_{n+1}^{\infty} or R_{n} \leq T_{n} \leq\int_{n}^{\infty} ? Say for example, I want to estimate the error in using the sum of the first 10 terms to approximate the sum of the series. The textbook seems to use...

I have been given the series sum 1/sqrt(3n-2) from n=1 to infinity I am supposed to use the limit comparison test, and the comparison series my book suggests is sum 1/sqrt n from n=1 to infinity, which I know is a divergent p series However, when I take the limit of one divided by the...

Homework Statement is the series convergent or divergent. Sum from 2 to ∞ of 1/ [n(ln n)^.5] Homework Equations comparison test? The Attempt at a Solution Is it possible to use the comparison test for this problem. Also, could I compare this with something like: (1/[(ln n)^.5]...

Homework Statement \sum\limits_{n = 2}^\infty {\frac{{\sqrt n }}{{n - 1}}} Homework Equations The Attempt at a Solution \begin{array}{l} a_n = \frac{{\sqrt n }}{{n - 1}},\,\,b_n = \frac{{\sqrt n }}{n} = \frac{{n^{1/2} }}{n} = \frac{1}{{n^{1/2} }}\,\,{\rm{is_ a_ p - series_...