By definition we have $\sum a_n + \sum b_n = \sum a_n + b_n$ If two series on RHS converge then clearly $\sum a_n + b_n = A+B$
But, Does this definition hold when series are conditionally convergent? Or at least one of them is conditionally convergent?
I couldn't find a direct example but consider this
let $a_n = x_n y_n$ and $b_n = c_nd_n$ now suppose $a_n$ and $b_n$ to converge conditionally. Then in that case, we can let partial sums of $x_n$ and $c_n$ to be bounded and $y_n$ and $d_n$ to monotonically decrease to $0$
Is it impossible to find the corresponding sequences so that $\sum a_n + b_n $ has partial sums that are unbounded - thus diverge. ?