# Thread: any example of sequences that their series satisfy these conditions

1. ## any example of sequences that their series satisfy these conditions

$\sum{x_{n}^{2}}$ and $\sum{y_{n}^{2}}$ converge. But:

$\sum{(x_{n}+y_{n})^{2}}$ does not converge.

2. $\sum (x_n+y_n)^2$ converges if and only if $\sum x_n^2+y_n^2 + 2x_ny_n$ converges if and only if $\sum 2x_ny_n$.

But $|2x_ny_n| \leq x_n^2+y_n^2$.
And $\sum x_n^2+y_n^2$ converges.