My Math Forum Using the Binomial Theorem to write a power series

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November 13th, 2012, 01:52 PM   #1
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Using the Binomial Theorem to write a power series

Hello!

Using the Binomial Theorem write [attachment=1:3ibjj3gt]gif.latex.gif[/attachment:3ibjj3gt] as a power series.

I got

[attachment=0:3ibjj3gt]gif.latex1.gif[/attachment:3ibjj3gt]

Is this correct? And how can i find the radius of convergence for this?
Attached Images
 gif.latex.gif (400 Bytes, 162 views) gif.latex1.gif (1.2 KB, 162 views)

 November 13th, 2012, 04:58 PM #2 Math Team   Joined: Sep 2007 Posts: 2,409 Thanks: 6 Re: Using the Binomial Theorem to write a power series I haven't done the detailed calculations but it certainly looks correct. And to find the radius of convergence use the "ratio test". $\frac{1(3)(5)\cdot\cdot\cdot(2(n+1)-1)x^{4(n+1)}}{2^{n+1}(n+1)!}\frac{2^nn!}{1(3)(5)\c dot\cdot\cdot(2n-1)x^{4n}}= \frac{1(3)(5)\cdot\cdot\cdot(2n- 2)}{1(3)(5)\cdot\cdot\cdot(2n-1)}\frac{2^n}{2^{n+1}}\frac{n!}{(n+1)!}x^{4}= \frac{1}{2n-1}\frac{1}{2}\frac{1}{n+1}x^4$ In order that the sequence converge, the limit of that must be less than 1. Find the limit, set it less than 1, and determine what x must be.
November 13th, 2012, 08:24 PM   #3
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Re: Using the Binomial Theorem to write a power series

Thank you very much!
I have one more question.
How can I use the results from this to find a power series representation for

[attachment=0:1gz724ny]gif.latex.gif[/attachment:1gz724ny]
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 gif.latex.gif (465 Bytes, 139 views)

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