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## Homework Statement

For what real values of x does this series converge?

[tex]\sum_{n=1}^{∞}{\frac{1}{(\frac{1}{x})^n + x^n}}[/tex]

## The Attempt at a Solution

I've rewritten the series as [tex]\sum_{n=1}^{∞}{\frac{x^n}{1 + x^{2n}}}[/tex] and I know that I can make each term larger, say

[tex]{\frac{x^n}{1 + x^{2n}}} < {\frac{x^n}{x^{2n}}} = \left({\frac{x}{x^{2}}}\right)^n = \left(\frac{1}{x}\right)^n[/tex]

Then this will converge, by comparison to the geometric series, when |x| > 1. But surely this can't be a method for finding ALL values of x for which the original series converges, can it? Because I changed the series , so all I've found is the values of x for which [tex]\sum_{n=1}^{∞}{\frac{x^n}{x^{2n}}}[/tex] converges. How can I go about this in a different way?

Then there's the question of uniform convergence. Can I just use the M-test for both parts, and say that

[tex]\left|\frac{x^n}{1 + x^{2n}}\right| < \left|\frac{x^n}{x^{2n}}\right| = \left|\frac{1}{x}\right|^n = M_n[/tex]

so, [tex]\sum_{n=1}^{∞}{M_n} = \sum_{n=1}^{∞} \left|\frac{1}{x}\right|^n[/tex] converges uniformly when |x| > 1?

I just fear that I'm simplifying things too much and not getting ALL of the possible values of x such that the series converges. Any suggestions?

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