Results 1 to 12 of 12

Math Help - Calculus I: (multi) infinite sums

  1. #1
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7

    Calculus I: (multi) infinite sums

    Another, well, interesting but not necessarily "challenging" problem from the series of problems that I started posting a few days ago. I hope you'll like it!

    Suppose the integer n \geq 1 is given. Evaluate I_n=\sum_{j_1 = 1}^{\infty} \sum_{j_2 = 1}^{\infty} \cdots \sum_{j_n = 1}^{\infty} \frac{1}{j_1 j_2 \cdots j_n(j_1 + j_2 + \cdots + j_n)}.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie bakerconspiracy's Avatar
    Joined
    Mar 2009
    Posts
    20
    What is this, N Sums from 1 to infinity of \frac{1}{j^{n}(nj)}=\frac{1}{nj^{n+1}}?

    What is the context of this problem? What are you trying to do with it; Since j and n are in N, this is a subseries of \frac{1}{j} , so I believe this will diverge no matter what. (could be wrong here)

    Almost looks like the end of an induction proof of sorts. Let me know if I can help you work on this, looks fun.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by bakerconspiracy View Post
    What is this, N Sums from 1 to infinity of \frac{1}{j^{n}(nj)}=\frac{1}{nj^{n+1}}?

    What is the context of this problem? What are you trying to do with it; Since j and n are in N, this is a subseries of \frac{1}{j} , so I believe this will diverge no matter what. (could be wrong here)

    Almost looks like the end of an induction proof of sorts. Let me know if I can help you work on this, looks fun.
    Since this is the Challenging Problems and Puzzles subforum (where questions are posted as a challenge for other members rather than to get help with) I think you'll find that NonCommAlg already knows the solution to this question ....
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    here's a hint: \frac{1}{j_1 + j_2 + \cdots + j_n}=\int_0^1 x^{j_1 + j_2 + \cdots + j_n - 1} \ dx.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Oct 2007
    From
    London / Cambridge
    Posts
    591
    is it \frac{\pi^{2n}}{6^n} ?

    Bobak
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Senior Member bkarpuz's Avatar
    Joined
    Sep 2008
    From
    R
    Posts
    481
    Thanks
    2
    Quote Originally Posted by bakerconspiracy View Post
    What is this, N Sums from 1 to infinity of \frac{1}{j^{n}(nj)}=\frac{1}{nj^{n+1}}?

    What is the context of this problem? What are you trying to do with it; Since j and n are in N, this is a subseries of \frac{1}{j} , so I believe this will diverge no matter what. (could be wrong here)

    Almost looks like the end of an induction proof of sorts. Let me know if I can help you work on this, looks fun.
    this is really funny
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by bobak View Post

    is it \frac{\pi^{2n}}{6^n} ?

    Bobak
    no it's not! the answer is n! \zeta(n+1).

    solution: starting with the hint i gave you:

    I_n=\int_0^1 \frac{1}{x} \sum_{j_1=1}^{\infty} \sum_{j_2=1}^{\infty} \cdots \sum_{j_n=1}^{\infty} \frac{x^{j_1 + j_2 + \cdots + j_n}}{j_1 j_2 \cdots j_n} \ dx=\int_0^1 \frac{1}{x} \sum_{j_1=1}^{\infty}\frac{x^{j_1}}{j_1} \sum_{j_2=1}^{\infty} \frac{x^{j_2}}{j_2} \cdots \sum_{j_n=1}^{\infty} \frac{x^{j_n}}{j_n} \ dx

    =\int_0^1 \frac{1}{x} \left(\sum_{j=1}^{\infty} \frac{x^j}{j} \right)^n \ dx= \int_0^1 (-1)^n \frac{(\ln(1-x))^n}{x} \ dx=\int_0^{\infty} \frac{t^n e^{-t}}{1 - e^{-t}} \ dt [here we did the substitution: \ln(1-x) = -t]

    =\int_0^{\infty}t^n e^{-t} \sum_{k=0}^{\infty}e^{-kt} \ dt = \sum_{k=0}^{\infty} \int_0^{\infty}t^ne^{-(k+1)t} \ dt=\sum_{k=0}^{\infty}\frac{n!}{(k+1)^{n+1}}=n! \zeta(n+1).
    Follow Math Help Forum on Facebook and Google+

  8. #8
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    Calc I ? What university did you go to Sheesh! lol.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by o_O View Post

    Calc I ? What university did you go to Sheesh! lol.
    haha ... i agree the problem is not straightforward but you really don't need to know anything beyond Calc I (well, a good Calc I course i mean), right?
    Follow Math Help Forum on Facebook and Google+

  10. #10
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    Haha guess that says a lot about the education at the university I go to. At the very least, this would be Calc III material and that's without the recognition of the zeta function !

    Guess SFU is that far ahead, huh?
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by o_O View Post

    Guess SFU is that far ahead, huh?
    if you mean average calculus students, not at all! but there are some first year students in here who, with a little push, can easily solve this problem for at least n = 2.

    by the way, i didn't do my undergraduate at SFU (or anywhere in Canada).
    Follow Math Help Forum on Facebook and Google+

  12. #12
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    Quote Originally Posted by NonCommAlg View Post
    by the way, i didn't do my undergraduate at SFU (or anywhere in Canada).
    !?!?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Infinite Sums
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 25th 2010, 11:18 AM
  2. Replies: 2
    Last Post: April 9th 2010, 03:16 AM
  3. Multi-variable Calculus Problems
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 10th 2010, 12:37 PM
  4. Evaluating Infinite Sums
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 10th 2010, 09:53 AM
  5. Replies: 2
    Last Post: November 21st 2009, 12:51 AM

Search Tags


/mathhelpforum @mathhelpforum