Infinite series with all negative terms

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The discussion centers on the convergence of infinite series with negative terms, specifically addressing the power series ∑(-1)n-1(xn/n). It highlights a paradox where traditional convergence tests typically require positive terms, yet the series converges for -1<x≤1, even when evaluating at -0.45 yields all negative terms. The conclusion drawn is that a series can indeed converge despite having all negative terms, as demonstrated by the example of -e. This challenges conventional understanding and emphasizes the need to consider specific series behavior in convergence tests. Overall, negative terms do not preclude convergence in infinite series.
Ibraheem
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Hello,

I have been reviewing my textbook lately, and I came across a rather paradoxical statements. all of the convergence tests in my book state that the terms of the series has to be positive. However, when I solved this power series ∑(-1)n-1(xn/n), I found that it converges for -1<x≤1, but when I plugged -0.45 to test if the infinite series will converge, I got all the terms negative. So my question is can a series converge if all its terms are negative?
 
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Yes. E.g. ##-e##.
 

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