Calc BC(2?) Series Question

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March 15th 2012, 03:16 PM
Xeritas

Calc BC(2?) Series Question

My high school teacher gave us the following problem:

The coefficients of the power series (starting at n=0 and continuing for all real numbers) a_n(x-2)ⁿ satisfy a₀ = 5 and a_n= ((2n+1)/(3n+1))a_n-1 for all n≥1. What is the radius of convergence of the series?

I've been taught to use the "ratio" test. Effectively placing the n+1 term over the nth term, simplifying, and then evaluating. Based on my work I believe that the answer should be 0, but am unsure whether it is correct due to that fact that I was absent when the material was taught.

P.S. Does anyone know how to write sigma notation in a word processor or in html?
March 15th 2012, 03:43 PM
skeeter

Re: Calc BC(2?) Series Question

Quote:

Originally Posted by Xeritas

My high school teacher gave us the following problem:

The coefficients of the power series (starting at n=0 and continuing for all real numbers) a_n(x-2)ⁿ satisfy a₀ = 5 and a_n= ((2n+1)/(3n+1))a_n-1 for all n≥1. What is the radius of convergence of the series?

I've been taught to use the "ratio" test. Effectively placing the n+1 term over the nth term, simplifying, and then evaluating. Based on my work I believe that the answer should be 0, but am unsure whether it is correct due to that fact that I was absent when the material was taught.

P.S. Does anyone know how to write sigma notation in a word processor or in html?

note that the ratio test can also be done using the ratio of the nth term to the (n-1)st term ...

$\lim_{n \to \infty} \left|\frac{a_n(x-2)^n}{a_{n-1}(x-2)^{n-1}}\right| < 1$

$\lim_{n \to \infty} \left|\frac{2n+1}{3n+1} \cdot (x-2)\right| < 1$

$|x-2| \lim_{n \to \infty} \frac{2n+1}{3n+1} < 1$

$|x-2| \cdot \frac{2}{3} < 1$

$|x-2| < \frac{3}{2}$
March 15th 2012, 03:56 PM
Xeritas

Re: Calc BC(2?) Series Question

Thanks, I didn't think of that.