To simplify the complex rational expression $\frac{\frac{1}{a}+\frac{1}{b}}{\frac{1}{a^2}-\frac{1}{b^2}}$ by writing it as division, proceed as follows:

Step 1. Simplify the numerator and denominator. In the numerator, the common denominator is $ab$: rewrite as $\frac{b}{ab} + \frac{a}{ab} = \frac{b + a}{ab}$. In the denominator, the common denominator is $a^2b^2$: rewrite as $\frac{b^2}{a^2b^2} - \frac{a^2}{a^2b^2} = \frac{b^2 - a^2}{a^2b^2}$. This simplifies the overall structure to $\frac{\frac{b + a}{ab}}{\frac{b^2 - a^2}{a^2b^2}}$.

Step 2. Rewrite as division. Replace the main fraction bar with a division sign: $\frac{b + a}{ab} \div \frac{b^2 - a^2}{a^2b^2}$.

Step 3. Multiply and factor. Multiply by the reciprocal: $\frac{b + a}{ab} \cdot \frac{a^2b^2}{b^2 - a^2}$. Factor the difference of squares in the denominator to reveal common terms: $b^2 - a^2 = (b - a)(b + a)$. The expression expands to $\frac{(b + a) \cdot a^2b^2}{ab(b - a)(b + a)}$.

Step 4. Simplify. Divide out the common factors of $ab$ and $(b + a)$ (recognizing that $b + a$ is equivalent to $a + b$). The remaining simplified expression is $\frac{ab}{b - a}$.