True or False. Justify your answers.

$($a$)$ If $\backslash$sum$\_($n$=7)^(\backslash$infty $)$ x$\_($n$)$ converges then $\backslash$sum$\_($n$=0)^(\backslash$infty $)$ x$\_($n$)$ also converges.

$($b$)$ If $\{$x$\_($n$)\}$ converges to $0$ then the sequence of partial sums $\{$s$\_($k$)\}$ also converges.

$($c$)$ If $\backslash$sum x$\_($n$)$ converges then $\backslash$sum $2$x$\_($n$)$ also converges.

$($d$)$ If $\backslash$sum x$\_($n$)$ is convergent and $\backslash$sum y$\_($n$)$ is divergent then $\backslash$sum x$\_($n$)+$y$\_($n$)$ is divergent.

$($e$)$ If $\backslash$sum x$\_($n$)$ and $\backslash$sum y$\_($n$)$ are both divergent series then $\backslash$sum x$\_($n$)+$y$\_($n$)$ also diverges.