How do solve the following recurrence?
$$ T(n) = \frac{1}{2} T\left(\frac{n}{2}\right) + \frac{1}{n}. $$
Master's theorem cannot be applied as $a$ is equal to 0.5 which is less than 1. Hence the theorem fails. How do I solve a recurrence when the theorem fails?
You can explicitly unroll the recursion: $$ \begin{align*} T(n) &= \frac{1}{n} + \frac{1}{2} T\left(\frac{n}{2}\right) \\ &= \frac{1}{n} + \frac{1}{n} + \frac{1}{4} T\left(\frac{n}{4}\right) \\ &= \cdots \\ &= \frac{m}{n} + \frac{1}{2^m} T \left(\frac{n}{2^m}\right). \end{align*} $$ This easily leads to the solution $$T(n) = \frac{\log_2 n + T(1)}{n},$$ which is valid for powers of 2.