# Polynomial sum of negligible functions need not be negligible

Let $$\{\epsilon_i\}_{n \in \mathbb{N}}$$ be a sequence of negligible functions and $$q(n)$$ be a polynomial in $$n$$. Then $$f(n) = \sum_{i = 1}^{q(n)} \epsilon_i(n)$$ need not be a negligible function.

Ideas

A typical negligible function is $$2^{-n}$$. Maybe we can expand it to the family $$2^{-(n+i)}$$ and set $$q(n)$$ to something that will give a sufficient large sum [didn't work]

Consider the negligible functions $$\epsilon_i(n) = \begin{cases} 1 & \text{if } n \le i \\ 0 & \text{if } n > i \\ \end{cases}$$
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...