# Thread: Power Series - Convergence and Sum

1. ## Power Series - Convergence and Sum

Consider power series

$\sum a_{n}x^n = 1 + 2x + 3x^2 + x^3 + 2x^4 + 3x^5 + x^6 ...$

in which coefficients an = 1, 2, 3, 1, 2, 3, 1 ... are periodic of period 3. Find the radius of convergence and the sum of this power series.

For the radius, is it simply a task of doing the limit test, where $\lim \frac{a_{n+1}x^{n+1}}{a_{n}x^n}$?

2. Originally Posted by Lord Darkin
Consider power series

$\sum a_{n}x^n = 1 + 2x + 3x^2 + x^3 + 2x^4 + 3x^5 + x^6 ...$

in which coefficients an = 1, 2, 3, 1, 2, 3, 1 ... are periodic of period 3. Find the radius of convergence and the sum of this power series.

For the radius, is it simply a task of doing the limit test, where $\lim \frac{a_{n+1}x^{n+1}}{a_{n}x^n}$?
Note that for positive $x$ we have that $\sum_{n\in\mathbb{N}}a_nx^n\leqslant 3\sum_{n\in\mathbb{N}}x^n$

3. How do I apply what you said into this problem?

4. Originally Posted by Lord Darkin
How do I apply what you said into this problem?
Figure it out

5. $\lim a_{n+1} * x < 1$

$\lim x < \frac{1}{a_{n+1}}$

$x = \frac{1}{3}, \frac{3}{2}, 2$