Results 1 to 9 of 9

Math Help - Series: Absolutely convergent, con convergent or divergent

  1. #1
    Junior Member
    Joined
    Feb 2009
    Posts
    44

    Series: Absolutely convergent, con convergent or divergent

    The series from 1 to infinity of n!/n^n? I tried the ratio and root test and they didn't work, unless I messed up. I was thinking a comparison test, but I'm not sure if you can verify something is absolutely convergent using the comparison test or limit comparison test.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,
    Quote Originally Posted by thehollow89 View Post
    The series from 1 to infinity of n!/n^n? I tried the ratio and root test and they didn't work, unless I messed up. I was thinking a comparison test, but I'm not sure if you can verify something is absolutely convergent using the comparison test or limit comparison test.
    The term in the series is positive, so here, absolute convergence and convergence are equivalent.
    Use the ratio test :
    a_n=\frac{n!}{n^n}

    \frac{a_{n+1}}{a_n}=\frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!}=\frac{(n+1) n^n}{(n+1)^n (n+1)}=\frac{n^n}{(n+1)^n}
    Now you have to take the limit as n goes to infinity :
    \frac{n^n}{(n+1)^n}=\left(\frac{n}{n+1}\right)^n=\  left(\frac{n+1}{n}\right)^{-n}=\left(1+\frac 1n\right)^{-n}

    now use the fact that \lim_{n \to \infty} \left(1+\frac 1n\right)^n=e>1
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2009
    Posts
    44
    So it also goes to e if the nth power is going to negative infinity? I was thinking something similar, but my notes are probably wrong I'm now thinking.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by thehollow89 View Post
    So it also goes to e if the nth power is going to negative infinity? I was thinking something similar, but my notes are probably wrong I'm now thinking.
    \left(1+\frac 1n\right)^{-n} is just another way for writing \frac{1}{\left(1+\frac 1n\right)^n}
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Feb 2009
    Posts
    44
    Wouldn't that make it 1/e then?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by thehollow89 View Post
    Wouldn't that make it 1/e then?
    Yes it does... I never said it didn't
    But with knowing these limits, you can say whether the ratio test fails or not.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Feb 2009
    Posts
    44
    Quote Originally Posted by Moo View Post
    Yes it does... I never said it didn't
    But with knowing these limits, you can say whether the ratio test fails or not.
    Okay just making sure. Thank you for your help, I hate making dumb mistakes. It's normal in Math to learn from mistakes right? lol
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by thehollow89 View Post
    Okay just making sure. Thank you for your help, I hate making dumb mistakes. It's normal in Math to learn from mistakes right? lol
    Yes, of course ! But there is no mistake if you don't try ^^
    Next time, it would be better if you post what you would have done so far
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Feb 2009
    Posts
    44
    Quote Originally Posted by Moo View Post
    Yes, of course ! But there is no mistake if you don't try ^^
    Next time, it would be better if you post what you would have done so far
    I will next time, I think it'd be better if I learn some LaTeX? I think that's what people use to post the Math symbols and what not.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Series Absolutely Convergent, xn/(1+xn^2) absolutely convergent
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: November 6th 2011, 12:36 PM
  2. Replies: 2
    Last Post: May 2nd 2010, 03:25 AM
  3. Replies: 1
    Last Post: April 25th 2010, 08:46 PM
  4. Replies: 2
    Last Post: August 4th 2009, 01:05 PM
  5. Replies: 3
    Last Post: April 6th 2009, 10:03 PM

Search Tags


/mathhelpforum @mathhelpforum