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- Thread starter Neon32
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If the series converges for all ##x## then by definition the radius of convergence is ##\infty##.

- #3

FactChecker

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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.I've 2 questions

1) Why do we take absolute of the power series?

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.2) I don't get why the interval of convergence is from -inifinity to +infinity.

- #4

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That is called the ratio test of convergence. (https://en.wikipedia.org/wiki/Ratio_test) The absolute value

I've seen different problems with the ratio test and they didn't use absolute but when it comes to power series, they use it. Why?

- #5

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I've seen different problems with the ratio test and they didn't use absolute but when it comes to power series, they use it. Why?

If you don't take the absolute value and the series has either alternating terms or at least infinitely many positive and negative terms, then the limit won't exist - unless it's 0. The test is better with the absolute value.

- #6

Mark44

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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}##.I've seen different problems with the ratio test and they didn't use absolute but when it comes to power series, they use it. Why?

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