# What is Comparison test: Definition and 105 Discussions

In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series.

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2. ### Prove limit comparison test for Integrals

Attempt: Note we must have that ## f>0 ## and ## g>0 ## from some place or ## f<0 ## and ## g<0 ## from some place or ## g ,f ## have the same sign in ## [ 1, +\infty) ##. Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...
3. ### Using comparison tests and limit comparison test

The answer sheet states that the series converges by limit comparison test (the second way). In the case of this particular problem, would it be also okay to use the comparison test, as shown above? (The first way) Thank you!
4. ### MHB Does the Comparison Test Determine Convergence or Divergence of Series?

Use the comparison test to determine if the series series convergences or divergences $$S_{6}=\sum_{n=1}^{\infty} \dfrac{1}{n^2 \ln{n} -10}$$ ok if i follow the example given the next step alegedly would be... $$\dfrac{1}{n^2 \ln{n} -10}<\dfrac{1}{n^2 \ln{n}}$$ $\tiny{242 UHM}$
5. ### MHB First Comparison Test for Series .... Sohrab Theorem 2.3.9 ....

I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition). I am focused on Chapter 2: Sequences and Series of Real Numbers ... ... I need help with the proof of Theorem 2.3.9 (a) ... Theorem 2.3.9 reads as follows: Now, we can prove Theorem 2.3.9 (a) using the Cauchy...
6. ### Comparison test of infinite series

Homework Statement Homework EquationsThe Attempt at a Solution So the book is saying that this series diverges, i have learned my lesson and have stopped doubting the authors of this book but i don't understand how this series diverges. ok i can use the comparison test using 1/3n and 1/3n...
7. ### Does this series converge? Using the limit comparison test

The problem In this problem I am supposed to show that the following series converges by somehow comparing it to ## \frac{1}{k\sqrt{k}} ## : $$\sum^{\infty}_{k=1} \left( \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} \right)$$ The attempt ## \frac{1}{\sqrt{k}} - \frac{1}{\sqrt{k+1}} =...
8. ### Series Comparison Test for Divergence: sin(1/n) vs 1/(1+n)

Homework Statement ##\sum _{n=0}^{\infty }\:\sin \left(\frac{1}{n}\right)## Homework Equations The Attempt at a Solution Can I try comparison test by ##\left(\frac{1}{1+n}\right)<sin\left(\frac{1}{n}\right)## since ##\left(\frac{1}{1+n}\right)## diverges also...
9. ### Proving the convergence of series

Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...
10. ### Why does (-1)^n(sin(pi/n)) converge when (sin(p/n)) diverges

Homework Statement I know that ∑n=1 to infinity (sin(p/n)) diverges due using comparison test with pi/n, despite it approaching 0 as n approaches infinity. However, an alternating series with (-1)^n*sin(pi/n) converges. Which does not make sense because it consists of two diverging functions...
11. ### Elliptic functions proof -- convergence series on lattice

Homework Statement Hi I am looking at the proof attached for the theorem attached that: If ##s \in R##, then ##\sum'_{w\in\Omega} |w|^-s ## converges iff ##s > 2## where ##\Omega \in C## is a lattice with basis ##{w_1,w_2}##. For any integer ##r \geq 0 ## : ##\Omega_r := {mw_1+nw_2|m,n \in...
12. ### Comparison test for series convergence (trig function)

Homework Statement Use a comparison test to determine whether this series converges: \sum_{x=1}^{\infty }\sin ^2(\frac{1}{x}) Homework EquationsThe Attempt at a Solution At small values of x: \sin x\approx x a_{x}=\sin \frac{1}{x} b_{x}=\frac{1}{x} \lim...
13. ### Use comparison test to see if series converges

Homework Statement \sum_{x=2}^{\infty } \frac{1}{(lnx)^9} Homework EquationsThe Attempt at a Solution x \geqslant 2 0 \leqslant lnx < x 0 < \frac{1}{x} < \frac{1}{lnx} From this we know that 1 / lnx diverges and I wanted to use this fact to show that 1 / [(lnx) ^ 9] diverges but at k...
14. ### Compare & Determine: The 1/n! Series Convergence/Divergence

Homework Statement Determine whether the series converges or diverges. ∞ ∑ 1/n! n=1 Homework Equations If ∑bn is convergent and an≤bn for all n, then ∑an is also convergent. Suppose that ∑an and ∑bn are series with positive terms. If lim an = C n→∞ bn where c is finite number and c>o...
15. ### MHB Limit Comparison Test: Does L Approaching Infinity Matter?

The limit comparison test states that if $a_n$ and $b_n$ are both positive and $L = \lim_{{n}\to{\infty} } \frac{a_n}{b_n} > 0$ then $\sum_{}^{} a_n$ will converge if $\sum_{}^{} b_n$ and $\sum_{}^{} a_n$ will diverge if $\sum_{}^{} b_n$ diverges. Does this rule also apply if $L$ diverges to...
16. ### MHB Determining the convergence or divergence of a sequence using comparison test

I have this series: $$\sum_{k = 1}^{\infty} {4}^{\frac{1}{k}}$$ To solve this, I am trying to compare it to this series $$\sum_{k = 1}^{\infty} {4}^{k}$$ So, I can let $a_k = {4}^{\frac{1}{k}}$ and $b_k = {4}^{k}$ These seem to be both positive series and $0 \le a_k \le b_k$ Therefore...
17. ### I How does the limit comparison test for integrability go?

Hi everybody! I have another question about integrability, especially about the limit comparison test. The script my teacher wrote states: (roughly translated from German) Limit test: Let -∞ < a < b ≤ ∞ and the functions f: [a,b) → [0,∞) and f: [a,b) → (0,∞) be proper integrable for any c ∈...
18. ### MHB Series Convergence with Comparison Test

Hey, I am working on Calculus III and Analysis, I really need help with this one problem. I am not even sure where to begin with this problem. I have attached my assignment to this thread and the problem I need help with is A. Thank you!
19. ### MHB Does the Series $$\sum_{n=1}^{\infty} \left[n(n+1)\right]^{-1/2}$$ Converge?

Use the comparison test to see if \sum_{1}^{\infty}{\left[n\left(n+1\right)\right]}^{-\frac{1}{2}} converges? I tried n+1 \gt n, \therefore n(n+1) \gt n^2 , \therefore {\left[n(n+1)\right]}^{\frac{1}{2}} \gt n, \therefore {\left[n(n+1)\right]}^{-\frac{1}{2}} \lt \frac{1}{n} - no conclusion...
20. ### Comparison Test for improper integral

Homework Statement use the comparison theorem to determine whether ∫ 0→1 (e^-x/√x) dx converges. Homework Equations I used ∫ 0 → 1 (1/√x) dx to compare with the integral above The Attempt at a Solution i found that ∫ 0 → 1 (1/√x) dx = 2 ( by substituting 0 for t and take the limit of the...

34. ### MHB Improper Integrals - Comparison Test

Hey, not too sure about what function i would compare this integral from 1 to infinity of (3x^3 -2)/(x^6 +2) dx. I also have to show that it converges. Thanks!
35. ### Series to compare to for comparison test

Homework Statement Does \sum_{n=1}^{\infty}a_n where a_n = \frac{(n+1)^{1/3}-n^{1/3}}{n} converge or diverge? Homework Equations The Attempt at a Solution The ratio test is inconclusive, as is the root test. The limit is equal to 0, but that doesn't say much. I've tried to find...
36. ### Use the Limit Comparison Test to determine the series' convergence?

Homework Statement Use the Limit Comparison Test to determine if the series converges or diverges: Ʃ (4/(7+4n(ln^2(n))) from n=1 to ∞. (The denominator, for clarity, in words is: seven plus 4n times the natural log squared of n.) Homework Equations Limit Comparison Test: Let Σa(n) be the...
37. ### Limit comparison test intuition

If we have two sequences and the ratio of their limit is greater than zero, why does this mean that they either both converge or diverge? I don't understand why the test works. Also, what about lim[(1/x)/(1/x^2)] = lim x = ∞? The series of 1/x^2 converges but series of 1/x diverges...
38. ### Convergence of Natural Log function with the limit comparison test

Homework Statement Determine whether Ʃ(n from 1 to infinity) ln(n)/n^3 converges or diverges using the limit comparison test. Homework Equations I must use the limit comparison test to solve this problem-not allowed to use other tests. The Attempt at a Solution I know that the...
39. ### Direct Comparison Test - Improper Integrals

1. Homework Statement [/b] Use the direct comparison test to show that the following are convergent: (a)\int_1^∞ \frac{cos x\,dx}{x^2} I don't know how to choose a smaller function that converges similar to the one above. The main problem is i don't know where to start. A simple...
40. ### Calculus II i don't understand the proof for the limit comparison test

would someone please care to reword this proof for me? http://en.wikipedia.org/wiki/Limit_comparison_test it talks about ε, which is not even defined and then n0, which is again not defined, what the hell are all these variables... I'm sure someone here could do a better job organizing...
41. ### MHB Improper integrals (Comparison Test)

Use the comparison test to find out whether or not the following improper integral exist(converge)? integral(upper bound:infinity lower bound:2) 1/(1-x^2) dx Here's my solution for 3),but I think something went wrong For all x>=2 0<=-(2-2x)<=-(1-x^2) that means: 0<=-1/(1-x^2)<=-1/(2-2x)...
42. ### MHB Improper integrals (Comparison Test)

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
43. ### LCT Limit Comparison Test for Improper Integrals

Learning about the Limit Comparison Test for Improper Integrals. I haven't gotten to any applications or actual problems yet. Just learning the theory so far, and have a question on the very beginning of it.Homework Statement f(x) ~ g(x) as x→a, then \frac{f(x)}{g(x)} = 1 (that is, f(x)...
44. ### Proving Convergence of Direct Comparison Test for \sum \frac{3}{n^{2} + 1}

Homework Statement Show that: \sum \frac{3}{n^{2} + 1} converges from n = 1 to ∞ Homework Equations If Ʃbn converges, and Ʃan < Ʃbn. Ʃan also converges. The Attempt at a Solution \sum \frac{1}{n^{2}} converges \sum \frac{3}{n^{2} + 1} = 3 * \sum \frac{1}{n^{2} + 1}...
45. ### Comparison Test problem with infinite series

Homework Statement I need to use the Comparison Test or the Limit Comparison Test to determine whether or not this series converges: ∑ sin(1/n^2) from 1 to ∞ Homework Equations Limit Comparison Test: Let {An} and {Bn} be positive sequences. Assume the following limit exists: L =...
46. ### How to Use Comparison Test for Convergence?

Homework Statement Is the series convergent or divergent? \sum_{n=0}^{\infty}{\frac{1}{\sqrt{n+1}}} Homework Equations I can use any test but wolfram alpha says that it is divergent by comparison test. The Attempt at a Solution How do I apply comparison test? I can compare it to: \sum _{...
47. ### Convergence of tan(1/n) using Direct Comparison Test

Homework Statement Use any test to determine whether the series converges.Homework Equations \displaystyle \sum^{∞}_{n=1} tan(1/n) The Attempt at a Solution Direct Comparison Test tan(1/n) > 1/n By integral test: 1/n diverges thus, by dct, tan(1/n) diverges.
48. ### Does the Limit Comparison Test Work for Divergent Integrals?

Homework Statement use limit comparison test. Homework Equations \displaystyle\int_2^∞ {\frac{1}{\sqrt{x^2 - 1}} dx} The Attempt at a Solution I have tried usin 1/x as the comparison function, but when applying the test it comes out to 0, not an L -> 0 < L < ∞
49. ### Convergence of Integral Using Limit and Direct Comparison Tests | Homework Help

Homework Statement Use direct comparison test or limit comparison test to determine if the integral converges.Homework Equations \displaystyle\int_0^6 {\frac{dx}{9-x^2}} The Attempt at a Solution If i were to use the limit comparison test, would these integrals fit the criteria. ** if the...
50. ### Can someone explain the direct comparison test to me in detail?

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...