Polynomial sum of negligible functions need not be negligible
Consider the negligible functions
$$\epsilon_i(n) = \begin{cases} 1 &amp; \text{if } n \le i \\ 0 &amp; \text{if } n &gt; i \\ \end{cases}$$

and $q(n) = n$

It should be easy to show that each $\epsilon_i$ function is, in fact, negligible, but the sum $f(n)$ is not...