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wisheskernel
 Post subject: The radius of convergence and interval of convergence
PostPosted: Wed, 18 Apr 2012 11:53:20 UTC 
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How to determine the radius of convergence and interval of convergence for the following power series?
$ \sum_{n=1}^{\infty} \frac{sin(nx)}{n^{2}}
Only jump right into the ratio test?
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outermeasure
 Post subject: Re: The radius of convergence and interval of convergence
PostPosted: Wed, 18 Apr 2012 12:34:38 UTC 
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wisheskernel wrote:
How to determine the radius of convergence and interval of convergence for the following power series?
$ \sum_{n=1}^{\infty} \frac{sin(nx)}{n^{2}}
Only jump right into the ratio test?


Over \mathbb{C}? Over \mathbb{R}?

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Justin
 Post subject: Re: The radius of convergence and interval of convergence
PostPosted: Wed, 18 Apr 2012 18:20:10 UTC 
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wisheskernel wrote:
How to determine the radius of convergence and interval of convergence for the following power series?
$ \sum_{n=1}^{\infty} \frac{sin(nx)}{n^{2}}
Only jump right into the ratio test?


http://en.wikipedia.org/wiki/Weierstrass_M_test

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

"In great mathematics there is a very high degree of unexpectedness, combined with inevitability and economy."
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Shadow
 Post subject: Re: The radius of convergence and interval of convergence
PostPosted: Wed, 18 Apr 2012 21:08:57 UTC 
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Justin wrote:
wisheskernel wrote:
How to determine the radius of convergence and interval of convergence for the following power series?
$ \sum_{n=1}^{\infty} \frac{sin(nx)}{n^{2}}
Only jump right into the ratio test?


http://en.wikipedia.org/wiki/Weierstrass_M_test


In \mathbb{C} sine is not bounded...

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Justin
 Post subject: Re: The radius of convergence and interval of convergence
PostPosted: Wed, 18 Apr 2012 22:39:01 UTC 
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Shadow wrote:

In \mathbb{C} sine is not bounded...


I assumed (hopefully correctly) that the OP was referring to \mathbb{R}.

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"Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another." Bernard Le Bovier Fontenelle (1657-1757)

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G.H. Hardy (1877-1947)
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Shadow
 Post subject: Re: The radius of convergence and interval of convergence
PostPosted: Wed, 18 Apr 2012 23:18:37 UTC 
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Justin wrote:
Shadow wrote:

In \mathbb{C} sine is not bounded, and on \mathbb{R} this is just dominated convergence.


I assumed (hopefully correctly) that the OP was referring to \mathbb{R}.


It wasn't clear to me that you meant for the set to be \mathbb{R}, but yes that works in that case.

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wisheskernel
 Post subject: Re: The radius of convergence and interval of convergence
PostPosted: Thu, 19 Apr 2012 08:15:57 UTC 
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outermeasure wrote:
wisheskernel wrote:
How to determine the radius of convergence and interval of convergence for the following power series?
$ \sum_{n=1}^{\infty} \frac{sin(nx)}{n^{2}}
Only jump right into the ratio test?


Over \mathbb{C}? Over \mathbb{R}?


Over $\mathbb{R}, How to determine the radius and interval of convergence for the series?
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Shadow
 Post subject: Re: The radius of convergence and interval of convergence
PostPosted: Thu, 19 Apr 2012 16:14:45 UTC 
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wisheskernel wrote:
outermeasure wrote:
wisheskernel wrote:
How to determine the radius of convergence and interval of convergence for the following power series?
$ \sum_{n=1}^{\infty} \frac{sin(nx)}{n^{2}}
Only jump right into the ratio test?


Over \mathbb{C}? Over \mathbb{R}?


Over $\mathbb{R}, How to determine the radius and interval of convergence for the series?


The M-test is a good choice if you want this as a functional sum. If x is fixed, then you can just use the normal comparison test.

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