$\sum_{n=1}^{\infty}(A_n+B_n)=\sum_{n=1}^{\infty}A_n+\sum_{n=1}^{\infty}B_n $

Question

If $\sum_{n=1}^{\infty}A_n$ is convergent and $\sum_{n=1}^{\infty}B_n$ is divergent then can we write $\sum_{n=1}^{\infty}(A_n+B_n)=\sum_{n=1}^{\infty}A_n+\sum_{n=1}^{\infty}B_n $?

Answer 1

If $\sum_{n=1}^\infty B_n$ is divergent, it is not clear what is meant by $\sum_{n=1}^\infty B_n$. It might be $-\infty$, $+\infty$ or maybe even something else (consider, for example, $\sum_{n=1}^\infty(-1)^n$).

Answer 2

If at any two of the series $\sum_{n=1}^{\infty}(A_n+B_n)$, $\sum_{n=1}^{\infty}A_n$, $\sum_{n=1}^{\infty}B_n$ converge, then so does the third and $$ \sum_{n=1}^{\infty}(A_n+B_n)=\sum_{n=1}^{\infty}A_n+\sum_{n=1}^{\infty}B_n $$ So in your case, we can only conclude that $\sum_{n=1}^{\infty}(A_n+B_n)$ diverges. Normally, when two series both diverge, we do not say that they are equal. (Nor do we say that they are unequal.)

Answer 3

The statement is false in the extended real line, by example suppose that $A_n:=n$ and $B_n:=(-1)^n$, then $\sum_{n\geqslant 0}(A_n+B_n)=\infty $ however $\sum_{n\geqslant 0}A_n+\sum_{n\geqslant 0}B_n$ is not defined as $\sum_{n\geqslant 0}B_n$ doesnt converge to something in the extended real line, therefore the equality would be false.

In the standard real line when both sides of an equality doesn't exists then the equality doesn't make sense, that is, the statement is as meaningless as if you write 172345g1j3kg123hu1263rt. You "can" write it (that is, you already had written it!!!) but it have no mathematical meaning, in short: it is not mathematics.