For which values of x does this series converge?

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Homework Statement


For what real values of x does this series converge?

\sum_{n=1}^{∞}{\frac{1}{(\frac{1}{x})^n + x^n}}

The Attempt at a Solution



I've rewritten the series as \sum_{n=1}^{∞}{\frac{x^n}{1 + x^{2n}}} and I know that I can make each term larger, say
{\frac{x^n}{1 + x^{2n}}} < {\frac{x^n}{x^{2n}}} = \left({\frac{x}{x^{2}}}\right)^n = \left(\frac{1}{x}\right)^n

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 \sum_{n=1}^{∞}{\frac{x^n}{x^{2n}}} 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

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

so, \sum_{n=1}^{∞}{M_n} = \sum_{n=1}^{∞} \left|\frac{1}{x}\right|^n 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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this problem might be well suited to a comparison

say 0<x<1, then for all n
<br /> \frac{1}{x^n} + x^n &gt; \frac{1}{x^n}<br />

Hence
<br /> \frac{1}{\frac{1}{x^n} + x^n}&lt; \frac{1}{\frac{1}{x^n}}<br />

note you could do similar thing for x>1 and negative values
 
Okay, that makes sense. But since I'm only comparing to other series, can I be sure that I'm catching all of the values of x where the original series converges?

EDIT: Nevermind...since I'm comparing the original series based on specific intervals of x, I'll be fine.

So I'll get the same interval for convergence and uniform convergence by the M-test?
 
not quite

consider x=1/2, using the fact below do you think the series will converge?
<br /> \frac{1}{\frac{1}{(\frac{1}{2})^n} + (\frac{1}{2})^n}&lt; \frac{1}{\frac{1}{(\frac{1}{2})^n}}= \frac{1}{2^n}<br />


now consider x=2, using the fact below do you think the series will converge?
<br /> \frac{1}{\frac{1}{(2)^n} + (2)^n}&lt; \frac{1}{2^n}<br />

note the symmetry in arguments, for any x>1, you can use the same argument above for y=1/x, with 0<y<1
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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