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. ?
Best Answer
Let $(A_n)$ and $(B_n)$ be the sequences of partial sums for these series: $$ A_n:=\sum_{k=0}^{n}a_k,\qquad B_n:=\sum_{k=0}^{n}b_k. $$ By assumption, you know that $A_n\to A$ and $B_n\to B$ as $n\to\infty$, for some $A,B\in\mathbb{R}$.
Now, the sequence of partial sums for the combined series is $(C_n)$, where $$ C_n:=\sum_{k=0}^{n}(a_k+b_k) $$ Notice, though, that $$ C_n=\sum_{k=0}^{n}a_k+\sum_{k=0}^{n}b_k=A_n+B_n. $$ This is true whether the series converges conditionally or absolutely; why? Because at this point, we are only adding finitely many terms. But, the usual limit laws tell us that because $A_n$ and $B_n$ converge, we also have that $C_n$ converges and $\lim C_n=\lim A_n+\lim B_n$.
So yes, no matter what, the combined series does converge and to the limit you expect.