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Math Help - Freaky series using limit comparison test!

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    Freaky series using limit comparison test!

    How in the world do you determine the following series' convergence/divergence specifically using the limit comparison test?

    from n=1 to n=infinity

    ln (2n+1)/ (n^2 + 2n)



    I'm bamboozled and befuddled!
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    Quote Originally Posted by Kaitosan View Post
    How in the world do you determine the following series' convergence/divergence specifically using the limit comparison test?

    from n=1 to n=infinity

    ln (2n+1)/ (n^2 + 2n)
    I would write the n'th term as \frac{\ln(2n+1)}{n^2+2n} = \frac{\ln(2n+1)}{\sqrt n}\frac{\sqrt n}{n^2+2n}. Then because "ln(n) goes to infinity slower than any positive power of n", it follows that the first fraction becomes small. The second fraction, \frac{\sqrt n}{n^2+2n}, is approximately n^{-3/2}. So do the limit comparison test, comparing the given series with the convergent series \sum n^{-3/2}.
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    Hm. I understand things better now. Thanks a lot.
    Last edited by Kaitosan; March 10th 2009 at 10:22 AM.
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    Quote Originally Posted by Opalg View Post
    I would write the n'th term as \frac{\ln(2n+1)}{n^2+2n} = \frac{\ln(2n+1)}{\sqrt n}\frac{\sqrt n}{n^2+2n}. Then because "ln(n) goes to infinity slower than any positive power of n", it follows that the first fraction becomes small. The second fraction, \frac{\sqrt n}{n^2+2n}, is approximately n^{-3/2}. So do the limit comparison test, comparing the given series with the convergent series \sum n^{-3/2}.
    Also, what's to prevent me from using, say, "n" instead of sqrt(n)? In that way, it will appear that the series diverges....

    I know it's against the rule to bump topics but I don't care, I really need help. I might as well post another topic.

    Anyways, I followed Opalag's directions but, strangely, I got zero. According to the limit comparison test, if the number equals zero or infinity, then the result is inconclusive. Please clarify?

    Also, why would I not use like n^2 instead of sqrt(n)? If I had done that then the series appears to diverge!

    Please help. Thanks a lot!
    Last edited by Krizalid; March 11th 2009 at 11:35 AM.
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    Quote Originally Posted by Kaitosan View Post
    I know it's against the rule to bump topics but I don't care, I really need help. I might as well post another topic.

    Anyways, I followed Opalg's directions but, strangely, I got zero. According to the limit comparison test, if the number equals zero or infinity, then the result is inconclusive. Please clarify?

    Also, why would I not use like n^2 instead of sqrt(n)? If I had done that then the series appears to diverge!

    Please help. Thanks a lot!
    Okay, what the limit comparison test says is that if \textstyle\sum a_n and \textstyle\sum b_n are series of positive terms, \lim_{n\to\infty}\frac{a_n}{b_n} exists and \textstyle\sum b_n converges, then \textstyle\sum a_n converges. (The test is sometimes stated in the form that if the limit exists and is nonzero then \textstyle\sum a_n converges if and only if \textstyle\sum b_n converges, but that is not what is needed for this example.)

    If a_n = \frac{\ln(2n+1)}{n^2+2n}, and you choose b_n = 1/n^p for some p between and 2, then \textstyle\sum b_n will converge (because p>1), and \lim_{n\to\infty}\frac{a_n}{b_n} will be 0 (because p<2).

    I hope that makes it clearer. (If you get an infraction for bumping, tell the Moderator that I think you were justified on this occasion.)
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    Or we can bound the general term: we have \ln x< 2\sqrt{x}-2,\,\forall\,x>1, thus for n\ge1 it's \frac{\ln (2n+1)}{n^{2}+2n}< \frac{2\sqrt{2n+1}-2}{n^{2}+2n}< \frac{4\sqrt{n}}{n^{2}}=\frac{4}{n^{3/2}}, then your series converges since \sum\limits_{n=1}^{\infty }{\frac{1}{n^{3/2}}}<\infty .
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    Quote Originally Posted by Opalg View Post
    Okay, what the limit comparison test says is that if \textstyle\sum a_n and \textstyle\sum b_n are series of positive terms, \lim_{n\to\infty}\frac{a_n}{b_n} exists and \textstyle\sum b_n converges, then \textstyle\sum a_n converges. (The test is sometimes stated in the form that if the limit exists and is nonzero then \textstyle\sum a_n converges if and only if \textstyle\sum b_n converges, but that is not what is needed for this example.)

    If a_n = \frac{\ln(2n+1)}{n^2+2n}, and you choose b_n = 1/n^p for some p between and 2, then \textstyle\sum b_n will converge (because p>1), and \lim_{n\to\infty}\frac{a_n}{b_n} will be 0 (because p<2).

    I hope that makes it clearer. (If you get an infraction for bumping, tell the Moderator that I think you were justified on this occasion.)


    Why must p be between 1 and 2? You simplified the series so as to exclude the ln expression in order to dig up the right-handed series for the b series. In that process of simplification, you can input any number in the bottom left and and the top right, and each number brings a different convergence/divergence result!

    Also, you seems to contradict yourself. If an/bn is nonzero and finite and if bn converges, then an converges. But you says an/bn converges to zero!

    Let me repeat my understanding for the limit comparison test-



    To evaluate the series an, pick bn based on an's highest ordered term in each numerator and denominator. Then structure an/bn and look for the limit. If the limit is between zero and infinity and if bn converges, then an/bn converges. But an/bn converges to zero! I'm so confused.

    Please don't make me bump this topic again.
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    Quote Originally Posted by Kaitosan View Post
    Why must p be between 1 and 2? You simplified the series so as to exclude the ln expression in order to dig up the right-handed series for the b series. In that process of simplification, you can input any number in the bottom left and and the top right, and each number brings a different convergence/divergence result!

    Also, you seems to contradict yourself. If an/bn is nonzero and finite and if bn converges, then an converges. But you says an/bn converges to zero!
    Please read what I said. I stated very carefully the form of comparison test that I was using. I said that if the ratio a_n/b_n converges (to any limit, including the possibility that the limit might be zero) and if \textstyle\sum b_n converges, then \textstyle\sum a_n converges.

    If you take a_n = \frac{\ln(2n+1)}{n^2+2n} and b_n = 1/n^{3/2}, then \frac{a_n}{b_n} = \frac{n^{3/2}\ln(2n+1)}{n^2+2n} = \frac{\ln(2n+1)}{n^{1/2} + 2n^{-1/2}} \to 0 as n\to\infty. therefore, by the limit comparison test in the form that I stated, \textstyle\sum a_n converges.
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    Quote Originally Posted by Opalg View Post
    Please read what I said. I stated very carefully the form of comparison test that I was using. I said that if the ratio a_n/b_n converges (to any limit, including the possibility that the limit might be zero) and if \textstyle\sum b_n converges, then \textstyle\sum a_n converges.
    Looks like we found our "communication hitch." Based on what it looks like, there are two possibilities. First, my book is wrong. Or, secondly, you may be slightly off the definition of the limit comparison (I say this in humility, there's always a good chance that I'm wrong).

    My book gives the definition of the limit comparison test in the following -

    "Let an and bn be positive-termed series. Let L = lim (n to infinity) an/bn. If 0 < L < +infinity, then an and bn either both converge or both diverge. If L = 0 or L = +infinity, then the test is inconclusive."

    The issue here is the domain of the limit comparison test and whether it includes zero. It's there, I copied exactly as my book puts it. What's going on?

    I really appreciate the fact that you replied speedily!
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    Quote Originally Posted by Kaitosan View Post
    My book gives the definition of the limit comparison test in the following -

    "Let an and bn be positive-termed series. Let L = lim (n to infinity) an/bn. If 0 < L < +infinity, then an and bn either both converge or both diverge. If L = 0 or L = +infinity, then the test is inconclusive."

    The issue here is the domain of the limit comparison test and whether it includes zero. It's there, I copied exactly as my book puts it. What's going on?
    I can only quote what I said before:

    Quote Originally Posted by Opalg View Post
    Okay, what the limit comparison test says is that if \textstyle\sum a_n and \textstyle\sum b_n are series of positive terms, \lim_{n\to\infty}\frac{a_n}{b_n} exists and \textstyle\sum b_n converges, then \textstyle\sum a_n converges. (The test is sometimes stated in the form that if the limit exists and is nonzero then \color{red}\textstyle\sum a_n converges if and only if \color{red}\textstyle\sum b_n converges, but that is not what is needed for this example.)
    Your book uses the version in red, but the slightly different version that I was using is used by some other authors, and is more useful for dealing with the example in this thread.

    In fact, I think that your book is somewhat misleading in claiming that the test is inconclusive when the limit is zero or infinity. As my version of the test shows, it is possible to draw conclusions in these cases. However, in those situations the test can only be used to get an implication in one direction; in these cases it is not an "if and only if" test.
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    Aaaah! I see! Thank you so much, I understand. I'll keep in mind that there are two "versions" of the limit comparison test then. Thanks for not giving me up.
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