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Math Help - Need help for convergence/integrals..

  1. #1
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    Question Need help for convergence/integrals..

    Hello i really need some help with these:

    1. Find out if they are convergent - and if so find the SUM.


    2. Find the absolute/partial convergence or divergence:


    3. Find the interval of convergence:


    4. Find and draw the limit of the function:


    5. Find out if the function has a limit and if so find the limit:


    6. Find R, represent the double integral as iterated integral in two ways (vertical simple and horizontal simple):0


    7. Find the particular solution:


    Thanks in advance
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Only seven problems!

    A little (very little help):

    \displaystyle\lim_{k \to{+}\infty}{\dfrac{1}{\sqrt[k]{e}}}=1\neq 0

    so,

    \displaystyle\sum_{k=1}^{+\infty}\dfrac{1}{\sqrt[k]{e}}

    is divergent.

    Fernando Revilla
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Another one:

    \displaystyle\lim_{k \to{+}\infty} \left(1+\dfrac{1}{2k+1}\right)^k=\ldots=\sqrt{e}\n  eq 0

    so, the corresponding series is divergent.

    Fernando Revilla

    P.S. What have you tried for the rest?. This should not be a monologue.
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  4. #4
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    sorry Fernando - My solutions for the first problem are the same
    for the second problem - not much but i transformed it in
    and stopped there. Now i'm on the third problem trying to solve it..
    thanks for the help so far

    ------ just noticed k -> infinity ------ it shoud be k -> 1 -----
    Last edited by karagorge; December 15th 2010 at 09:04 AM. Reason: mistake
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  5. #5
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    Just solved the third one:
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by karagorge View Post
    sorry Fernando - My solutions for the first problem are the same
    for the second problem - not much but i transformed it in
    Take into account that \sin k\pi=0 so, the series

    \displaystyle\sum_{k=1}^{+\infty} \dfrac{\sin k\pi}{2}=\displaystyle\sum_{k=1}^{+\infty}0

    is convergent with sum 0 .

    Fernando Revilla
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  7. #7
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by karagorge View Post
    Just solved the third one:
    That is right. Now, the series is absolutely convergent on (-11/2,11/2) . You need to study the convergence at the end points of the interval.

    (i) For x=11/2 you'll obtain:

    \dfrac{1}{22}\displaystyle\sum_{k=1}^{+\infty}\dfr  ac{1}{n}

    divergent (harmonic series and algebra of series)

    (ii) For x=11/2 you'll obtain:

    \dfrac{1}{22}\displaystyle\sum_{k=1}^{+\infty}\dfr  ac{(-1)^n}{n}

    conditionally convergent (alternating harmonic series and algebra of series).

    So, the interval of convergence is [-11/2,11/2) .

    Fernando Revilla
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  8. #8
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by karagorge View Post
    4. Find and draw the limit of the function:
    When (x,y)\rightarrow \textrm{?}

    Fernando Revilla
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  9. #9
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    Aaaa don't know - that might be a mistake - i'll ask tomorrow.
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  10. #10
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by karagorge View Post
    Aaaa don't know - that might be a mistake - i'll ask tomorrow.
    All right. Don't forget to show some work at every problem.

    Fernando Revilla
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  11. #11
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by karagorge View Post
    5. Find out if the function has a limit and if so find the limit:
     x^4-y^4 = (x^2+y^2)(x^2-y^2) What does this tell you?

    7. Find the particular solution:
    Assuming you meant  3y''+8y'-3y = 0 ...

    We must first solve  3\lambda^2+8\lambda-3 = 0 . This gives us  \displaystyle \lambda = \{-3,\tfrac13\} .

    Therefore  y = c_1e^{-3x}+c_2e^{x/3} . Now use your conditions to solve for  c_1, c_2 .
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  12. #12
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    Thanks for the fifth one - but i really can't figure out the 7th - Please finish it.. At the moment i'm solving other math problems so please forgive me for not answering on time or not including often in the process.
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  13. #13
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    I just solved the 6th one:
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