Prove that this particular series, $\sum_{n=1}^\infty\frac{nx^2(cos(nx^2))}{e^n(x^2+1)}$ converges uniformly and absolutely in $\mathbf{R}$.
I can think of how to construct other series that are larger than this series and yet convergent, which would prove convergence by comparison. But I do not know a rigorous proof for uniform and absolute convergence. If someone could provide a thorough proof for both, that would be very helpful for my understanding.
Since $$\left|\frac{nx^2(\cos(nx^2))}{e^n(x^2+1)}\right|\leq \frac{n}{e^n}<\frac{6}{n^2}$$ the claim follows from Weierstrass M-test.