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Given two infinite series $A(x):=\sum_na_n(x)$, and $B(x):=\sum_nb_n(x)$, suppose they have radius of convergence $r_1$ and $r_2$ respectively, then what is the radius of convergence of $\sum_n a_n(x)+b_n(x)$?

Is the region of convergence the intersection of the region of convergence of $A(x)$ and $B(x)$? I doubt that this is true by the ratio test.

z.z
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By the ratio test, there is an $m$ sufficiently large and there are $r_a,r_b<1$ such that forall $n\ge m$,

$$|a_n|\le |a_m|r_a^{n-m}$$ and

$$|b_n|\le |b_m|r_b^{n-m}.$$

Then

$$|a_n+b_n|\le|a_n|+|b_n|\le |a_m|r_a^{n-m}+|b_m|r_b^{n-m}\le(|a_m|+|b_m|)(\max(r_a,r_b))^{n-m},$$

assuming $r_a\ge r_b$.