I know that [tex]\sum[/tex] of tan(1/x^2) converges, but why?

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I know that [tex]\sum[/tex] of tan(1/x^2) converges, but why?
 

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  • #2
mathman
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I presume x is supposed to be an integer, but you should specify domain.
In any case, for large x, tan(1/x2) ~ 1/x2, which implies convergence (assuming x starts > 0).
 
  • #3
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For a more rigourous explanation, apply the inequalities

[tex]\sin\left(\frac{1}{x^2}\right)\leq \frac{1}{x^2}~\text{and}~1-\frac{1}{2x^4}\leq \cos\left(\frac{1}{x^2}\right)[/tex]

These inequalities follow immediately from regarding the Taylor series of sin and cos. This also gives you an inequality:

[tex]\tan\left(\frac{1}{x^2}\right)\leq\frac{\frac{1}{x^2}}{1-\frac{1}{2x^4}}[/tex]

Working this out should give you a nice upper bound of the tangent function. So the answer follows from the comparison test.
 
  • #4
Mute
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For a more rigourous explanation, apply the inequalities

[tex]\sin\left(\frac{1}{x^2}\right)\leq \frac{1}{x^2}~\text{and}~1-\frac{1}{2x^4}\leq \cos\left(\frac{1}{x^2}\right)[/tex]

These inequalities follow immediately from regarding the Taylor series of sin and cos. This also gives you an inequality:

[tex]\tan\left(\frac{1}{x^2}\right)\leq\frac{\frac{1}{x^2}}{1-\frac{1}{2x^4}}[/tex]

Working this out should give you a nice upper bound of the tangent function. So the answer follows from the comparison test.
You can't divide inequalities.

e.g., 1 < 1.5 and 0.25 < 0.5, but that doesn't mean that 1/0.25 < 1.5/0.5 => 4 < 3!
 
  • #5
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You can't divide inequalities.

e.g., 1 < 1.5 and 0.25 < 0.5, but that doesn't mean that 1/0.25 < 1.5/0.5 => 4 < 3!
Sure you can!! If 0<a<b and 0<c<d, then [tex]\frac{a}{d}<\frac{b}{c}[/tex].
In your example, that would be 1<1.5 and 0.25<0.5, then [tex]\frac{1}{0.5}<\frac{1.5}{0.25}[/tex], which is perfectly true.

You have to be careful WHICH PART of the inequality you divide though...
 
  • #6
Mute
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Sure you can!! If 0<a<b and 0<c<d, then [tex]\frac{a}{d}<\frac{b}{c}[/tex].
In your example, that would be 1<1.5 and 0.25<0.5, then [tex]\frac{1}{0.5}<\frac{1.5}{0.25}[/tex], which is perfectly true.
I stand corrected!

You have to be careful WHICH PART of the inequality you divide though...
I guess I wasn't careful enough!
 

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