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Math Help - Need help with these homework problems

  1. #1
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    Thumbs up Need help with these homework problems

    Hi All,

    I took my time to scan these 7 problems from the book to see if you guys can help me out with any of them. I've missed about a week of class due to work travel, and it would really help if you guys can help me with the solutions, as I have a test in 4 days and need to know how to do these.

    http://modernvirtual.com/1.png
    http://modernvirtual.com/2.png

    Thanks in advance!
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  2. #2
    Senior Member yeKciM's Avatar
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    let's go from the beginning

    1. problem

    what the criteria for testing convergence do you know? (if u know any try applying it and if u have problem say where )
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  3. #3
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    I honestly don't know even where to start with these. I've missed about five days of a 3 hour class, so that was a lot of material. I tried to look up the material online and tried wolfram alpha as well as trying a few methods by hand and still cannot even comprehend how to do these.
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  4. #4
    Senior Member yeKciM's Avatar
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    u have D'alamberth (sorry if i mistype his name)

    \displaystyle \sum_{n=1}^{\infty}a_n

    then :

    \displaystyle q=\lim_{n\to \infty} \frac {a_{n+1}}{a_n}

    so u have a cases :

    q=\begin{cases}<br />
q<1 & converges  \\ <br />
 q>1& diverges \\ <br />
 q=1& ?  <br />
\end{cases}

    or Rabel's (again sorry for mistyping)

    \displaystyle \sum_{n=1}^{\infty}a_n

    \displaystyle q=\lim_{n\to \infty} n[\frac {a_{n+1}}{a_n}-1]

    q=\begin{cases}<br />
q<1 & converges  \\ <br />
 q>1& diverges \\ <br />
 q=1& ?  <br />
\end{cases}

    or u can do it "by definition"
    definition say that series converges or diverges if

    \displaystyle \lim_{N\to \infty} S_N

    is number then converges
    if infinity nuber or doesn't exist it diverges

    (S_N)_{N\in\mathbb{N}} is array of partial sums
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  5. #5
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    Ok, that helped a ton. I think i have them all figured out except for #3 and #5. How would I go about those?
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  6. #6
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by nxd10 View Post
    Ok, that helped a ton. I think i have them all figured out except for #3 and #5. How would I go about those?
    first of all those criteria that I wrote there isn't only them.... there are few more

    as 3. problem i'm trying to google translate it (sorry if i understand what's point i'll post it )

    for 5. problem i think is something to do with "Štolc" theorem (i think is something like "Shtolch" or )

    he says that :

    "if (x_n)_{n\in\mathbb{N}} and (y_n)_{n\in\mathbb{N}} arbitrary real sequences such that a sequence (y_n) satisfies:


    1 (y_n) monotonically increasing

    2 \displaystyle \lim_{n\to\infty} y_n = \infty


    if exist :

     \displaystyle \lim_{n\to\infty} \frac {x_{n+1}-x_n}{y_{n+1}-y_n}

    then exist and

     \displaystyle \lim_{n\to\infty} \frac {x_n}{y_n}

    and they are equal

    i think that's it


    hehehe or u can (i think again ) for #5 just (to se if is increasing or decreasing) just put  a_n < a_{n+1} and if it's true then is increasing but if  a_n > a_{n+1} then is decreasing (we do it like that here )
    Last edited by yeKciM; August 4th 2010 at 12:28 PM.
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