A question is Given that $$\sum\frac{(-1)^nx^{2n}}{n^p(1+x^{2n})} $$ } for what value of $\;p\;$ given series of functions is uniformly convergence for all vales of $x\in \mathbb{R}$
Solution i tried- The question is simple ,if we apply $M_n-$ test then we can see that $$\left | \frac{(-1)^nx^{2n}}{n^p(1+x^{2n})} \right |\leq \frac{1}{n^p}$$ for $x \in \mathbb{R}$ and also here $\displaystyle M_n=\frac{1}{n^p}$ and $\sum M_n$ is convergent for for $p>1$
My Question- As per definition of $M_n$ test they only focus on $[a,b]$ but in this question it is asked on whole $\mathbb{R}$ ,is this legal to use $M_n$ test like i used in above question ,i am confused here because in my book they are using $M_n$ test Dini's test and Drichlit's test over $\mathbb{R}$ but nowhere in the definition of these tests they consider $ \mathbb{R}$.
Please help