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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]

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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...

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  • $\begingroup$ I would like to add that, using the limit definition will fail, since not all time, the summation of the limit is equal to limit of the sum. $\endgroup$
    – kelalaka
    Nov 17, 2018 at 14:37

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