# Convergence of series with non-negative vs positive terms

The number 0 is neither positive nor negative. There are series such as $$\sum_{n=1}^{\infty}\frac{\left|n-2\right|}{2^{\ n}}$$ where one of its terms is 0.

In convergence tests, the positive vs non-negative condition is used apparently interchangeably and differently depending on the source (e.g. https://en.wikipedia.org/wiki/Convergence_tests).

My (unconfirmed guess) is that positive is used to mean non-negative. However 0 is always permitted

Do you know any convergence test that does not accept the 0 term?

• The ratio test, for obvious reasons, is not applicable when there are infinitely many terms equal to $0$. Commented Aug 20, 2023 at 9:48
• For a sequence of non-negative terms, if you have some terms which vanish, then just ignore them. The convergence is unaffected: one series converges if and only if the other converges, and in this case converge to the same value. So, it is no loss of generality to assume everything is positive (and if every term vanishes, then you of course simply have the series is trivially equal to $0$). But anyway, the only time you need to be careful is when division is involved, and most tests do not involve division, so you’re fine. Commented Aug 20, 2023 at 9:51
• Positive is never used to mean non-negative (barring the occasional person who is careless in their word choices - a sin in mathematics). The very reason the ungainly term "non-negative" is used in mathematics is to handle situations where $0$ is allowed. However, that doesn't mean everyone quotes theorems with the weakest possible hypotheses. So the tests may be stated with the stronger hypothesis of positivity, instead of the weaker hypothesis of non-negativity. Commented Aug 21, 2023 at 17:07