Express $0.\overline{1}+0.\overline{01}+0.\overline{0001}$ as a common fraction.
This problem involves adding repeating decimals, which can be tricky. One way to approach this is to try to write each repeating decimal as a fraction using a clever trick. For example, to write $0.\overline{1}$ as a fraction, I can let $x=0.\overline{1}$ and then multiply both sides by 10 to get $10x=1.\overline{1}$. Then I can subtract $x$ from $10x$ to get rid of the decimal part and obtain $9x=1$. Solving for $x$, I get $x=\frac{1}{9}$, which is the fraction form of $0.\overline{1}$. Similarly, I can write $0.\overline{01}$ as a fraction by letting $y=0.\overline{01}$ and then multiplying both sides by 100 to get $100y=1.\overline{01}$. Subtracting $y$ from $100y$, I get $99y=1$, which implies $y=\frac{1}{99}$. Finally, I can write $0.\overline{0001}$ as a fraction by letting $z=0.\overline{0001}$ and then multiplying both sides by 10000 to get $10000z=1.\overline{0001}$. Subtracting $z$ from $10000z$, I get $9999z=1$, which implies $z=\frac{1}{9999}$. Now that I have the fraction forms of the repeating decimals, I can add them up using a common denominator. The least common multiple of 9, 99, and 9999 is 9999, so I can rewrite the fractions as $\frac{1111}{9999}+\frac{101}{9999}+\frac{1}{9999}$. Adding the numerators, I get $\frac{1213}{9999}$. This is the fraction form of the original sum of repeating decimals. 