<s>[INST] What method is most preferred to find nature of the series $\sum_{n=1}^{\infty} \frac{a^n}{a^n+x^n}$? I tried to use D'Alembert's ratio test to test the convergence of the given series $\sum\limits_{n=1}^{\infty}  \frac{a^n}{a^n+x^n}$ but I could only solve for $\frac{a}{x} <1$ and it is coming out to be converging but for $\frac{x}{a} <1$, I am not able to do it with ratio test. 
 [/INST] $$\frac{a^n}{a^n + x^n} = \frac{1}{1 + (\frac{x}{a})^n}$$
Ratio test:
$$\frac{\frac{a^{n+1}}{a^{n+1} + x^{n+1}}}{\frac{a^n}{a^n + x^n}} = a \frac{a^n + x^n}{a^{n+1} + x^{n+1}} =  \frac{1 + (\frac{x}{a})^n}{1 + (\frac{x}{a})^{n+1}}$$


*

*If $\vert \frac{x}{a} \vert \leq 1$, the above limits goes to $1$. And the ratio test here is inconclusive. Note here that $\frac{a^n}{a^n + x^n} = \frac{1}{1 + (\frac{x}{a})^n}$ does not go to zero (goes to $1$). The series diverges. This test is referred to as $n^{th}$ term test.

*If $\vert \frac{x}{a} \vert > 1$, the above limit acts as 
$$\frac{(\frac{x}{a})^n}{(\frac{x}{a})^{n+1}} = \frac{a}{x} < 1$$


Thanks @DanielWainfleet for the remark on absolute values. 
</s>