For which values of x does this series converge?

  • Thread starter moxy
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  • #1
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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?
 
Last edited:

Answers and Replies

  • #2
lanedance
Homework Helper
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2
this problem might be well suited to a comparison

say 0<x<1, then for all n
[tex]
\frac{1}{x^n} + x^n > \frac{1}{x^n}
[/tex]

Hence
[tex]
\frac{1}{\frac{1}{x^n} + x^n}< \frac{1}{\frac{1}{x^n}}
[/tex]

note you could do similar thing for x>1 and negative values
 
  • #3
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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?
 
  • #4
lanedance
Homework Helper
3,304
2
not quite

consider x=1/2, using the fact below do you think the series will converge?
[tex]
\frac{1}{\frac{1}{(\frac{1}{2})^n} + (\frac{1}{2})^n}< \frac{1}{\frac{1}{(\frac{1}{2})^n}}= \frac{1}{2^n}
[/tex]


now consider x=2, using the fact below do you think the series will converge?
[tex]
\frac{1}{\frac{1}{(2)^n} + (2)^n}< \frac{1}{2^n}
[/tex]

note the symmetry in arguments, for any x>1, you can use the same argument above for y=1/x, with 0<y<1
 

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