Example where lim(Xn + Yn) exists but doesn't equal lim Xn + lim Yn

[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 [itex]\sum (x_n+y_n)=\sum x_n + \sum y_n[/tex], the radii of converge need not be the same.[/itex]

 

[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]

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