Pointwise and uniform converges of a series

For $x \in \mathbb{R}$ observe the series:

$$\sum_{n=1}^{\infty} \frac{1}{n^2} cos\Big( \frac{x^3}{n} \Bigr)$$

Show that the series converges pointwise for $-\infty \lt x \lt \infty$ and that the series also converges uniform on all $\mathbb{R}$.

Would be sufficient to use Weierstrass M-test, to show that the series converges pointwise for $-\infty \lt x \lt \infty$. Since we know that it has a convergent majorant series. $$\sum_{n=1}^{\infty} \left|\frac{1}{n^2} cos\Big( \frac{x^3}{n} \Bigr)\right|\le \sum_{n=1}^{\infty} \frac{1}{n^2}$$

We know that the seres $\sum_{n=1}^{\infty} \frac{1}{n^2}$ converges and by Weierstrass M-test, this implies that the series converges uniform on $\mathbb{R}$ and this implies that the series converges pointwise or do I have to specifically argue for the pointwise convergence?

1 Answer

If a series (or sequence) converges uniformly to a function $f$, then it conveges pointwise to the same function. So, no, you don't have to specifically argue for the pointwise convergence.