Example of Xn + Yn Limit | Help with Problem

[Trigger]

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.

 

[Galileo]

By the old theorem:

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

Maybe also allowed: Even if \sum (x_n+y_n)=\sum x_n + \sum y_n[/tex], the radii of converge need not be the same.

 

[Hyperreality]

Does the two sequence Xn and Yn defined??

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

\sum_{n=1}^\infty\frac{1}{n}
is undefined, and also

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

but \sum_{n=1}^\infty\frac{1}{n} - \sum_{n=1}^\infty\frac{(-1)^nx^n}{n}=\gamma=0.577...

 

[Hyperreality]

Does the two sequence Xn and Yn defined??

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

\sum_{n=1}^\infty\frac{1}{n}
is undefined, and also

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

but \sum_{n=1}^\infty\frac{1}{n} - \sum_{n=1}^\infty\frac{(-1)^nx^n}{n}=\gamma=0.577...

 

[Tsss]

X_n=\sum_{k=1}^n 1
Y_n=\sum_{k=1}^n -1