Convergence radius of $\sum_{n=1}^\infty \frac{(4-x)^n}{\sqrt{n^4+5}}$

Question

Find the convergence radius of $$\sum_{n=1}^\infty \frac{(4-x)^n}{\sqrt{n^4+5}}$$

I've recently started self-learning about series, so I'm having a little trouble. Looking at this example, I tried the $n^{th}$-root test:

$$\sqrt[n]{\bigg|\frac{(4-x)^n}{\sqrt{n^4+5}}\bigg|}=\frac{|4-x|}{\sqrt[2n]{n^4+5}}\to0$$

Does this mean the sum converges for all $x$ and convergence radius is $\infty$?

Any help is appreciated

Answer 1

No. We have

$$ 1 \le \sqrt[n]{\sqrt{n^4+5}} \le \sqrt[n]{\sqrt{4n^4}}=\sqrt[n]{2n^2}$$

for $n \ge 1.$

Hence

$$\sqrt[n]{\sqrt{n^4+5}} \to 1$$

as $n \to \infty.$

Conclusion ?