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Math Help - convergence/divergence(urgent help needed plz!)

  1. #1
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    convergence/divergence(urgent help needed plz!)

    Use any method you can to decide the convergence of
    i)



    ii)



    iii)



    iv)
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by matty888 View Post
    Use any method you can to decide the convergence of
    i)
    Evaluate \lim_{n\to \infty}\frac{\sqrt{n}}{\ln n}

    ii)
    Evaluate \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n

    iii)
    It's a geometric series

    iv)
    Use the ratio test. (it's often useful with fractions involving factorials)
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  3. #3
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    please help

    Hello could you do the first one for me please im really stuck!thanks
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  4. #4
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    Quote Originally Posted by matty888 View Post
    Hello could you do the first one for me please im really stuck!thanks
    Do you realise that \sum a_n diverges if \lim_{n \rightarrow \infty} a_n \neq 0 ?
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  5. #5
    Member disclaimer's Avatar
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    Quote Originally Posted by mr fantastic View Post
    Do you realise that \sum a_n diverges if \lim_{n \rightarrow \infty} a_n \neq 0 ?
    \lim_{n\rightarrow{\infty}}\frac{\sqrt{n}}{\ln{n}} is obviously infinity, but how to prove it?
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  6. #6
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    Quote Originally Posted by disclaimer View Post
    \lim_{n\rightarrow{\infty}}\frac{\sqrt{n}}{\ln{n}} is obviously infinity, but how to prove it?
    Well L'Hopitals rule will work

    RonL
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  7. #7
    Member disclaimer's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Well L'Hopitals rule will work

    RonL
    I've never seen L'Hospital's rule used to calculate a limit of a sequence (sequences don't have derivatives).
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  8. #8
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    Quote Originally Posted by disclaimer View Post
    I've never seen L'Hospitals rule used to calculate a limit of a sequence (sequences don't have derivatives).


    What do you mean? Does it matter whether it is a limit of a sequence or limit of any other function
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  9. #9
    Grand Panjandrum
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    Quote Originally Posted by disclaimer View Post
    I've never seen L'Hospital's rule used to calculate a limit of a sequence (sequences don't have derivatives).
    Technically you are looking at the limit of \frac{\sqrt{x}}{\ln(x)} but the two limits are equal (as long as you go from the continuous to the discrete anyway). (If you are not convinced suppose otherwise and you will rapidly find a contradiction)

    RonL
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  10. #10
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    Quote Originally Posted by CaptainBlack View Post
    Technically you are looking at the limit of \frac{\sqrt{x}}{\ln(x)} but the two limits are equal (as long as you go from the continuous to the discrete anyway). (If you are not convinced suppose otherwise and you will rapidly find a contradiction)

    RonL
    When I was doing series, to be able to use L'Hopital's rule I would simply say:

    a_n= f(n)

    I found that my teacher accepted that, then I could use L'H because I was dealing with a continuous function.
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  11. #11
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by flyingsquirrel View Post
    Hi

    Evaluate \lim_{n\to \infty}\frac{\sqrt{n}}{\ln n}
    Evaluate \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n

    It's a geometric series

    Use the ratio test. (it's often useful with fractions involving factorials)
    This one \lim_{n \to {\infty}}\bigg(1+\frac{1}{n}\bigg)^{n} is the famous definiton for e\approx{2.71828183...}..which means the sum diverges by the n-th term test...if you want to see how this was derived(the limit) look here http://www.mathhelpforum.com/math-he...-tutorial.html
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  12. #12
    Moo
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    Quote Originally Posted by Mathstud28 View Post
    This one \lim_{n \to {\infty}}\bigg(1+\frac{1}{n}\bigg)^{n} is the famous definiton for e\approx{2.71828183.....which means the sum diverges by the n-th term test...if you want to see how this was derived(the limit) look here http://www.mathhelpforum.com/math-he...-tutorial.html
    Hello,

    I sincerely think that Flyingsquirrel knows how to do
    He didn't say he couldn't do it, he just gave the way to do ^^

    Are you paying the advertisings ?
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  13. #13
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Moo View Post
    Hello,

    I sincerely think that Flyingsquirrel knows how to do
    He didn't say he couldn't do it, he just gave the way to do ^^

    Are you paying the advertisings ?
    Oh...I know Flyingsqurriel can do it...I meant to quote the poster...
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