Answer
$$a_{n}=-1, \ \ \ \ b_{n}=\left(\frac{1}{2}\right)^{n}+1$$
Work Step by Step
Consider $$a_{n}=-1, \ \ \ \ b_{n}=\left(\frac{1}{2}\right)^{n}+1$$
Since
\begin{align*}
\lim_{n\to \infty } a_n &=-1\neq 0\\
\lim_{n\to \infty } b_n &= 1\neq 0
\end{align*}
Then $ \sum a_n,\ \ \sum b_n $ are divergent series, but
$$ \sum_{n=1}^{\infty} (a_n+b_n)=\sum_{n=1}^{\infty} \left(\frac{1}{2}\right)^{n} $$
which is a geometric series with $r= 1/2$ and has the sum
$$ \frac{a}{1-r}= \frac{\frac{1}{2}}{1-\frac{1}{2}}=1$$
