In a power series
$\sum c_n (x-a)^n$
What qualifies to be in the $(x-a)$ part?
eg.
- $2-x$
- $5x-2$
- $\frac{x-3}{2}$
- A combination of the above? $\frac{3-2x}{6}$
In a question I am working on, finding the radius of convergence of
$\sum_{n=0}^{\infty} \frac{(5-4x)^{2n}}{9^n n^{5/4}}$
the model answer has
$c_{2n} = \frac{4^{2n}}{9^n n^{5/4}}$
But what I actually did:
$\sum_{n=0}^{\infty} \frac{(5-4x)^{2n}}{9^n n^{5/4}} \\ =\frac{(-1)^{2n}(4x-5)^{2n}}{9^nn^{5/4}} \\ =\frac{(-1)^{2n}(x-\frac{5}{4})^{2n}}{4^{2n} 9^n n^{5/4}}$
So my $c_n=\frac{(-1)^{2n}}{4^{2n} 9^n n^{5/4}}$
Why isit in the given answer ($c_N$), its not an alternating series, and why is $4^{2n}$ in the numerator?
UPDATE: OK about the $4$ in denominator question, that was my mistake.