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That is called the ratio test of convergence. (https://en.wikipedia.org/wiki/Ratio_test) The absolute value is all you need to test convergence.Neon32 said:
It converges if the absolute value of the limit is less than 1. In this case, the limit of the ratio is 0 regardless of the value of x. So it converges for any value of x.
FactChecker said:
Neon32 said:
A power series is a series in powers of a variable such as x. The Ratio Test requires that all terms be positive, but with a variable, some terms could be negative, if x is negative. The problems you've seen that didn't use absolute values were almost certainly series in which all the terms were positive, such as ##\sum_{n = 1}^\infty \frac 1 {n^2 + 1}##.Neon32 said:
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