What is Integral test: Definition and 54 Discussions

In mathematics, the integral test for convergence is a method used to test infinite series of monotonous terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.

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  1. F

    Infinite Series (Integral Test)

    After evaluating the integral I found the following: (1/3)tan-1(e∞/3) = (1/3)tan-1(∞) = (1/3)(nπ/2), where n is an odd number. In this case I found multiple solutions to the problem. How do you prove it converges?
  2. F

    Infinite Series (Integral Test)

    I got the following expression: -(1/4)ln((n+2)/(n-2)) When I substitute "∞" in the expression I found it undefined. However, the book says the series converges. What am I doing wrong?
  3. N

    I Testing for Divergence using the Integral Test

    Hello all, I was working on some homework regarding testing for convergence and divergence of series and I was having trouble with a particular series (doesn't really matter which one) and tried almost all the methods; then tried the Integral Test, my series met the conditions of the...
  4. opus

    I Looking for help to understand the Integral Test

    Please see the attached images that reference the text. To my understanding, we wish to use the integral test to compare to a series to see if the series converges or diverges. In these two examples, we use ##\sum_{n=1}^\infty \frac{1}{n}## compared with ##\int_1^{k+1} \frac{1}{x}dx## and for...
  5. opus

    Does the series converge using the integral test?

    Homework Statement Use the integral test to compare the series to an appropriate improper integral, then use a comparison test to show the integral converges or diverges and conclude whether the initial series converges or diverges. ##\sum_{n=3}^\infty \frac{n^2+3}{n^{5/2}+n^2+n+1}## Homework...
  6. Y

    Integral Test: Analyzing $$\sum_{n=3}^{\infty} \frac{1}{n^2-4}$$

    Homework Statement Use the integral test to determine whether $$\sum_{n=3}^{\infty} \frac{1}{n^2-4}$$ converges or diverges.The Attempt at a Solution [/B] Taking the integral we have $$\int_{}^{\infty}\frac{dn}{n^2-4}$$ Note: Mary Baos text is having us write the integrals without lower bounds...
  7. 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.
  8. K

    Using the Integral Test to Show Sum is Less Than pi/2

    Homework Statement Use the integral test to show that the sum of the series gif.latex ##\sum_{n=1}^\infty \dfrac{1}{1+n^2}## is smaller than pi/2. Homework EquationsThe Attempt at a Solution I know that the series converges, and the integral converges to pi/4. As far as I´ve understood...
  9. karush

    MHB 242.tr.05 Use the integral test to determine if a series converges.

    $\tiny{242.tr.05}$ Use the integral test to determine if a series converges. $\displaystyle \sum_{n=1}^{\infty}\frac{1}{\sqrt{e^{2n}-1}}$ so... $\displaystyle \int_{1}^{\infty} \frac{1}{\sqrt{e^{2n}-1}}\, dn =\int_{1}^{\infty} (e^{2n}-1)^{1/2} \, dn $ so $u=e^{2n}-1\therefore du=2e^{2n}$
  10. Rectifier

    Is There a Constant Lower Bound for the Integral Test of Convergence?

    The problem I am trying to show that the following integral is convergent $$ \int^{\infty}_{2} \frac{1}{\sqrt{x^3-1}} \ dx $$The attempt ## x^3 - 1 \approx x^3 ## for ##x \rightarrow \infty##. Since ## x^3 -1 < x^3 ## there is this relation: ##\frac{1}{\sqrt{x^3-1}} > \frac{1}{\sqrt{x^3}}##...
  11. T

    B Integral test and its conclusion

    I'm really confused about this test. Suppose we let f(n)=an and f(x) follows all the conditions. When you take the integral of f(x) and gives you some value. What are you supposed to conclude from this value?
  12. samgrace

    I Cauchy's Integral Test for Convergence

    Hello, I am want to prove that: $$ \sum_{1}^{\infty} \frac{1}{n^{2} + 1} < \frac{1}{2} + \frac{1}{4}\pi $$ Cauchy's Convergence Integral If a function decreases as n tends to get large, say f(x), we can obtain decreasing functions of x, such that: $$ f(\nu - 1) \geqslant f(x) \geqslant...
  13. ognik

    MHB Is the series $ \sum_{n=2}^{\infty} \frac{1}{n^2} $ converges?

    Hi - just done the integral test on the Riemann zeta series, came out to $\frac{1}{p-1}$ I can clearly see it therefore converges for P > 1, is singular for p=1, but for p < 1 I can't see why it diverges? In the limit p < 1 just gets smaller? Would also like to check about p = 1, all I need...
  14. B

    MHB Integral Test: Analyzing 1/(n+1)^x with x > 1

    Hi, I am trying to use an integral test on the following series: The sum from n=0 to infinity on 1/(n+1)^x where x>1 I know the process of using the integral test however i am unsure as to how to evaluate the integral with the x in the series :/ Thanks in advance!
  15. P

    Integral Test of e^n/(e^(2n)+9)

    Homework Statement Using the integral test determine whether or not the following sum diverges. $$ \sum_{n=1}^{\infty} \frac{e^n}{e^{2n}+9} $$ Homework Equations The Attempt at a Solution $$ \sum_{n=1}^{\infty} \frac{e^n}{e^{2n}+9}=\sum_{n=1}^{\infty} \frac{e^n}{({e^{n})}^2+9} \\ \int^{\infty}...
  16. Pull and Twist

    MHB Remainder Estimate for Integral Test

    I'm working on the following problem and I have made it this far... am I on the correct path or am I doing this incorrectly?? I find series extremely confusing. Also... how do I find the error involved in the improved approximation? This is the series I am working with...
  17. Randall

    Use the integral test to determine if this series converges or diverges

    Homework Statement Use the integral test to determine if this series converges or diverges: sum from n=1 to infinity of n/(1+(n^2)) Homework Equations Integral test: a series and it's improper integral both either converge or both diverge The Attempt at a Solution see attached - I need help...
  18. RJLiberator

    Do You Adjust Bounds When Using the Integral Test for Series Convergence?

    Homework Statement When using the Integral Test do you need to change the bounds to n+1 and n-1 for an increasing and decreasing function respectively? This is a question that comes up when using the integral test. I think that you just use the original bounds for the integral. We are a bit...
  19. RJLiberator

    Root Test and Integral Test Question

    Homework Statement [/B] From K=4 to infinity the Σ (-1)^k (k/e^k) Converge or diverge? Use: a) Ratio Test b) Root Test c) Integral Test d) Alternating series test Homework EquationsThe Attempt at a Solution For the alternating series test and ratio test I have the correct answer that it...
  20. M

    Evaluating a limit by integral test

    Homework Statement Evaluate ## ∑^∞_{n=1} \frac {2}{n(n+2)} ## 2. The attempt at a solution I've solved this question simply enough by evaluating it as a telescoping series and found the answer as 3/2. Now, when applying the integral test, it only works when dealing with positive, decreasing...
  21. S

    MHB What is the result of integrating \frac{x}{x^2 + 1} and applying the limit?

    Suppose: \int^{\infty}_1 \frac{x}{x^2 + 1} \, dx I can't do this can I? Let u = x^2 + 1 so du = 2x \, dx \therefore \frac{du}{2} = x \, dx ?
  22. T

    Integral test - ultimately decreasing function

    My textbook states that "For the integral test to apply, it is not necessary that f be always decreasing. What is important is that f be ultimately decreasing." I'm just curious what is meant by this statement. How do you define a function that is "ultimately decreasing"? It'd be great if you...
  23. I

    Calculus II please explain integral test

    i understand the reason for if the evaluated integral converges then the series also converges. However, just because the evaluated integral diverges, why does this automatically mean that the related series also diverges? the integral consists of every number of the +x axis when evaluating...
  24. Y

    Question about the Integral Test

    So this is my first post, I was wondering can you explain the first two examples of this page? http://www.math.ubc.ca/~rathb/mar_6_p_4.jpg What I don't understand is why, if there is a horizontal asymptote at p = 0.99 < 1 on first example, it diverges to infinity but in the second...
  25. I

    Applying Integral Test: Why did they integrate from 1 to ∞ instead of

    Consider example 4 in the attachment. Why did they integrate from 1 to ∞ instead of e to ∞?
  26. B

    Determine If Integral Test Can Be Applied

    Homework Statement I attached the infinite series the question provided as a file. Homework Equations The Attempt at a Solution I deduced the general term to be ln(n)/n, so the infinite series would be written as \sum_{n=2}^{\infty} \frac{\ln(n)}{n} I took the derivative of the...
  27. B

    Geometric Series Test Rather Than Integral Test

    Homework Statement \sum_{n=1}^{\infty} \frac{1}{2^n} Homework Equations The Attempt at a Solution Could I some how manipulate this to fit a geometric series, so that I may instead use the geometric series test?
  28. 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?
  29. M

    Proof of Convergence by Integral Test and/or Comparison Test

    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...
  30. V

    Theory behind Integral Test with Riemann Sums

    Homework Statement I've seen two methods that prove the integral test for convergence, but I fear they contradict each other. Each method uses an improper integral where the function f(x) is positive, decreasing, and continuous and f(x) = an. What confuses me is one method starts off the...
  31. T

    Integral Test: What should I compare this series with to prove it's convergence?

    Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/HW%20Pictures/integraltest.PNG The Attempt at a Solution I know it is convergent because it is very similar to 1/n^1.5 which is convergent as well. However what would I compare this with using the Integral Test to...
  32. S

    Solving Series w/ Remainder Estimate & Integral Test

    It asks "use remainder estimate for integral test" to find series accurate to 3 dec? Homework Statement It says "Use the Remainder Estimate for the Integral Test to find the sum of the following series to three decimal places of accuracy." \sum^{\infty}_{n=1} \frac{1}{n^{3}} Homework...
  33. QuarkCharmer

    Did I Solve the Integral Test Correctly for This Question?

    Homework Statement Took a test today, one question I am not sure I got right. \int_{0}^{\infty} te^{-at}dt, when a>0 Homework Equations The Attempt at a Solution I set let infinity be b, then took the limit as b went to infinity of the integral with new bounds from 0 to b. My solution...
  34. M

    Using the integral test to test for divergence/convergence

    Homework Statement \sum_{n=1}^{\infty}\frac{8\arctan{n}}{1+n^2} Homework Equations The Attempt at a Solution so I am comparing it to the integral \int_{1}^{\infty}\frac{8\arctan{x}}{1+x^2} but at first i need to show that the function I am integrating is continuous, positive and...
  35. G

    Calculus II - Infinite Series - Integral Test Problem

    Homework Statement Determine whether the following series converges absolutely, converges conditionally or diverges. Show your work in applying any tests used. sigma[k=1,inf] [(-1)^k*k/sqrt(k^4+2)] Homework Equations integral csc(x) dx = -ln|sec(x)+cot(x)| + C csc(x) = 1/sin(x) sec(x) =...
  36. S

    Question about the integral test

    Homework Statement We have to determine whether \sum 1/n^2 + 4 is convergente or divergent Homework Equations I'm trying to work the problem through trigonometric substitution. I was wondering if I could just determine that by the P-series test, the function 1/n^2 will always be larger...
  37. V

    Question about the integral test

    Homework Statement (∞, n=1) ∑ ne-n Homework Equations The Attempt at a Solution (∞, n=1) ∑ ne-n limt--> ∞ ∫ xe-x dx from 1 to t here i tried to do integration by parts u = x du=dx dv= e-x dx v= -e-x not sure where to take limits at this point, is it like this limt--> ∞ -xe-x (from 1...
  38. P

    Improper integral of integral test

    Homework Statement determine the value of the improper integral when using the integral test to show that \sum k / e^k/5 is convergent. answers are given as a) 50/e b) -1 / 5e^1/5 c) 5 d) 5e e)1/50e The Attempt at a Solution f(x) = xe^-x/5 is continuos and positive for all...
  39. N

    Does the Integral Test \(\sum_{n=1}^{\infty}\frac{\ln n}{n^p}\) Converge?

    \sum_{n=1}^{\infty}\frac{lnn}{n^P} \int_{1}^{\infty}lnx(x^{-P})dx
  40. J

    Convergence: The Integral Test

    Homework Statement Is \sum \frac{1}{2n(2n+1)} convergent or divergent? (Note that the summation is from 1 to infinity) Homework Equations \int f(x) dx = L, (range is from 1 to infinity) IF L = \infty, divergent L < \infty, convergent. The Attempt at a Solution I...
  41. B

    Integral Test: Convergence/Divergence of $\sum_{x=2}^\infty (\ln x)^{-1}$

    Homework Statement Determine convergence or divergence using the integral test for. \sum _{x=2}^{\infty } \left( \ln \left( x \right) \right) ^{-1} Homework Equationsi should take the limit as b goes to infinity of \int _{2}^{b}\! \left( \ln \left( x \right) \right) ^{-1}{dx} The problem...
  42. C

    Convergence of Series Using Integral Test

    Homework Statement Use the integral test to determine the convergence or divergence of the series. \Sigma^{\infty}_{n=1}\frac{n^{k-1}}{n^{k}+c} k is a positive integer Homework Equations The Attempt at a Solution Consider: \int^{/infty}_{1}\frac{x^{k-1}}{x^{k}+c} dx Not...
  43. L

    Integral test for converging series

    Homework Statement In one of my problems, I'm asked to use integral test in order to determine whether the following function is converging or diverging: (summation from n=1 to inf) 1/(n^2-4n-5) I was just wondering how do we know if function is positive, continuous and decreasing because when...
  44. L

    Determine convergence of series using integral test

    Homework Statement How can i find if the following series is convergent or divergent using the INTEGRAL TEST? sigma (n=1 to infinity)= n/(n^4+1) Homework Equations The Attempt at a Solution The answer says that the initial step involves changing it to: 1/2(2x)/(1+(x^2)^2)..but...
  45. R

    Series Convergence: Integral Test and Homework Equations

    Homework Statement Determine whether this series converges or diverges : http://mathbin.net/equations/7548_0.png Homework Equations Integral Test The Attempt at a Solution http://mathbin.net/equations/7548_1.png from that it shows to diverge but book says it converges. What did I...
  46. R

    Does the Integral Test Result Indicate the Series Sum?

    Hi all, I just want to know a little something: When doing the integral test in order to find a sum, when might get a result (integral) of a certain number. As we know, getting a number as result an integral test means that this serie converges...but does that mean that the serie converges to...
  47. E

    Strange infinite series problem using integral test.

    Homework Statement I need to show that \Sigma\frac{1}{nlnnlnlnn} from n=27 to n=10^(100,000) is approximately equal to 8.1
  48. M

    Infinite series estimation using integral test

    Homework Statement Estimate \sum^{\infty}_{n=1}n^{-3/2} to within 0.01 Homework Equations \int^{\infty}_{n+1}f(x)dx\leq R_{n} \leq \int^{\infty}_{n}f(x)dx The Attempt at a Solution So my strategy was using the above formula to find Rn, where Rn = 0.01 or 1/10^2. Then that will give me the...
  49. H

    Integral test by comparison(Please verify my proof)

    Integral test by comparison(Please look at my work) Homework Statement Looking at the Integral a_n = \int_{0}^{\pi} \frac{sin(x)}{x+n\pi} prove that a_n \geq a_{n+1} Homework Equations The Attempt at a Solution Proof given the integral test of comparison and since...
  50. S

    Integral Test for Convergence/Divergence

    Use the integral test to determine convergence or divergence of the foloowing series: a) \infty (n=1) \sumn/n\hat{}2+1 b) \infty (n=2) \sum1/n\hat{}2+1