Power Series - Convergence and Sum

**Lord Darkin** · Mar 8th 2010, 02:10 PM

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

**Drexel28** · Mar 8th 2010, 02:56 PM

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 $\sum_{n\in\mathbb{N}}a_nx^n\leqslant 3\sum_{n\in\mathbb{N}}x^n$

**Lord Darkin** · Mar 8th 2010, 03:00 PM

How do I apply what you said into this problem?

**Drexel28** · Mar 8th 2010, 03:01 PM

Originally Posted by Lord Darkin

How do I apply what you said into this problem?

Figure it out

**Lord Darkin** · Mar 8th 2010, 03:19 PM

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

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

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

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