Find the properties of functional series

Question

I'd like some help finding the properties of the following series of powers:

$\sum_{k=1}^{\infty}\frac{3^\sqrt{k}}{k}x^{2k}$

I need the convergence area and the absolute convergence area.

Answer 1

Well, first of all we have a problem with the beginning of the sum. What happens when:

$$\lim_{\text{k}\to0}\frac{3^\sqrt{\text{k}}x^{2\text{k}}}{\text{k}}\tag1$$

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Now, when we have the following sum:

$$\sum_{\text{n}\ge1}\frac{3^\sqrt{\text{n}}x^{2\text{n}}}{\text{n}}\tag2$$

We can use the ratio test:

$$\lim_{\text{n}\to\infty}\left|\frac{\frac{3^\sqrt{\text{n}+1}x^{2\left(\text{n}+1\right)}}{\text{n}+1}}{\frac{3^\sqrt{\text{n}}x^{2\text{n}}}{\text{n}}}\right|=\left|x\right|^2\tag3$$

So $|x|<1$.