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Urgent Analysis Problem

  1. Dec 8, 2004 #1
    Please help with the following problem:
    Do not know where to start!

    Give an example of two sequences Xn(sum from 1 to infinity) and Yn(sum from 1 to infinity) where lim (as n tends to infinity) of (Xn + Yn ) exists but lim (as n goes to infinity) of(Xn +Yn) does not equal lim (as n goes to infinity) Xn + lim(as n goes to infinity) Yn.
     
  2. jcsd
  3. Dec 8, 2004 #2

    Galileo

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    By the old theorem:

    [tex]\sum_n (x_n+y_n)=\sum_n x_n + \sum_n y_n[/tex]
    if [itex] \sum x_n[/itex] and [itex]\sum_y_n[/itex] are convergent series, your only hope is to have either [itex] \sum x_n[/itex] or [itex] \sum y_n[/itex] divergent.

    Maybe also allowed: Even if [itex]\sum (x_n+y_n)=\sum x_n + \sum y_n[/tex], the radii of converge need not be the same.
     
  4. Dec 8, 2004 #3
    Does the two sequence Xn and Yn defined??

    If not (this is a wild guess), is gamma constant one? Because:

    [tex]\sum_{n=1}^\infty\frac{1}{n}[/tex]
    is undefined, and also

    [tex]-\sum_{n=1}^\infty\frac{(-1)^nx^n}{n}[/tex]
    is also undefined

    but [tex]\sum_{n=1}^\infty\frac{1}{n} - \sum_{n=1}^\infty\frac{(-1)^nx^n}{n}[/tex]=\gamma=0.577...
     
  5. Dec 8, 2004 #4
    Does the two sequence Xn and Yn defined??

    If not (this is a wild guess), is gamma constant one? Because:

    [tex]\sum_{n=1}^\infty\frac{1}{n}[/tex]
    is undefined, and also

    [tex]-\sum_{n=1}^\infty\frac{(-1)^nx^n}{n}[/tex]
    is also undefined

    but [tex]\sum_{n=1}^\infty\frac{1}{n} - \sum_{n=1}^\infty\frac{(-1)^nx^n}{n}=\gamma=0.577...[/tex]
     
  6. Dec 8, 2004 #5
    [tex]X_n=\sum_{k=1}^n 1[/tex]
    [tex]Y_n=\sum_{k=1}^n -1[/tex]
     
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