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appgolfer
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I know that [tex]\sum[/tex] of tan(1/x^2) converges, but why?
micromass said: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.
Mute said: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!
micromass said: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...
The sum of tan(1/x^2) converges because it satisfies the conditions of the Cauchy condensation test for convergence. This means that the series has a similar behavior to a geometric series, which is known to converge. Additionally, the function tan(1/x^2) decreases rapidly as x approaches infinity, ensuring convergence.
The convergence of the sum of tan(1/x^2) has important implications in the study of infinite series and the behavior of functions. It also has applications in calculus and in the analysis of complex systems.
The convergence of the sum of tan(1/x^2) can be proved using various convergence tests, such as the Cauchy condensation test, the comparison test, or the integral test. Each of these methods requires a different approach and level of mathematical knowledge.
No, the convergence of the sum of tan(1/x^2) is not limited to a specific range of values. As long as the series satisfies the appropriate conditions for convergence, it will converge for all values of x.
The convergence of the sum of tan(1/x^2) is not directly related to the convergence of other trigonometric series. However, some of the convergence tests used to prove the convergence of the sum of tan(1/x^2) can also be applied to other trigonometric series. Additionally, the techniques used to prove convergence may be similar for certain trigonometric series.