Question about Power Series

Question

I have this problem:

$$\sum_{n=0}^{\infty}\ 64^n\ (x-10)^{3n+1}$$

After using the Ratio Test I am left with this:

$$ \lim_{n\to\infty}\big|\frac{64^{n+1}(x-10)^{3n+4}}{64^n(x-10)^{3n+1}}\big| $$

When I simplify the problem... I get this

$$ \lim_{n\to\infty}\big|64(x-10)^3\big| $$

There's no "$n$" in the problem left... what would be the interval of convergence? I feel like I might've made a mistake somewhere or I missed something in class. Any help would be awesome, thanks!

Answer 1

No, you're perfectly correct. What that means is that $$\lim_{n\rightarrow \infty} |64(x-10)^3| = |64(x-10)^3|$$

So you can find the interval of convergence the way you normally do: finding those $x$ such that $|64(x-10)^3| < 1$.

Answer 2

Since you solved the problem, I just add for your curiosity that the summation has a very nicely surprising closed form $$\sum_{n=0}^{\infty}\ 64^n\ (x-10)^{3n+1}=\frac{x-10}{(41-4 x) \left(16 x^2-316 x+1561\right)}$$