Thread: Power Series - Convergence and Sum

Consider power series

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

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 $\displaystyle \lim \frac{a_{n+1}x^{n+1}}{a_{n}x^n}$?

Originally Posted by Lord Darkin

Consider power series

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

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 $\displaystyle \lim \frac{a_{n+1}x^{n+1}}{a_{n}x^n}$?

Note that for positive $\displaystyle x$ we have that $\displaystyle \sum_{n\in\mathbb{N}}a_nx^n\leqslant 3\sum_{n\in\mathbb{N}}x^n$

How do I apply what you said into this problem?

Originally Posted by Lord Darkin

How do I apply what you said into this problem?

Figure it out

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

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

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