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 a_{0} = 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?

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 a_{0} = 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 ...

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

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

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

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

$\displaystyle |x-2| < \frac{3}{2}$

Re: Calc BC(2?) Series Question

Thanks, I didn't think of that.