Thread: My calc book is very vague

1. My calc book is very vague

How do you find the convergence radius of a taylor polynomial

such as $\displaystyle \sum_{n=0}^infty\frac{n^2(x^(2n))}{3^(n-1)}$

2. If this doesn't work I give up

$\displaystyle \sum_{n=0}^{\infty}\frac{n^2x^{2n}}{3^{n-1}}\$

3. Originally Posted by Mathstud28
$\displaystyle \sum_{n=0}^{\infty}\frac{n^2x^{2n}}{3^{n-1}}\$

Try the ratio test.

$\displaystyle lim_{n \to \infty}\left| \frac{a_{n+1}}{a_n}\right|=L$

if $\displaystyle L < 0$ the series converges
if $\displaystyle L=1$ the test is inconclusive
if $\displaystyle L\ge 1$ The series diverges

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

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

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

So we want the series to converge so we want

$\displaystyle \left| \frac{x^2}{3}\right| < 1$

if we solve the inequality we get the radius of convergance.

Note you need to check the end points with another test.

4. Originally Posted by Mathstud28
$\displaystyle \sum_{n=0}^{\infty}\frac{n^2x^{2n}}{3^{n-1}}\$
Originally Posted by Mathstud28

5. So

Obviously we want $\displaystyle a_{n+1}\$<$\displaystyle a_{n}\$ because then that is what causes convergence?

6. Originally Posted by Mathstud28
Obviously we want $\displaystyle a_{n+1}\$<$\displaystyle a_{n}\$ because then that is what causes convergence?
no, we want $\displaystyle \lim_{n \to \infty} \left| \frac {a_{n + 1}}{a_n} \right| < 1$ ............then we have convergence by the ratio test

all you need to do is solve the last line TheEmptySet left you with for x. be sure to check the end-points as well

7. So

if its not too much to ask could you give me an example I work through it and you see if I am right? If thats too much work I understand...with all that LaTeX you have to go through to do it I wouldn't blame you haha