Prove Uniform Convergence on $\frac{cos(n\pi x))}{n}$ as $n \to \infty$

Question

Trying to prove uniform convergence and having a bit of difficulty working out $$ f_n(x)=\frac{cos(n\pi x)}{n}$$ as $n \to \infty$ on $x \in[0,1]$. I know that obviously because $n \to \infty$ then the denominator will be $\infty$ and the whole equation will tend $\to 0$.

However I am trying to prove whether or not this series of $f_n$ is uniformly convergent. So ultimately I am looking to figure out if there is an $x$ value which makes $$f_n(x) \not= 0 $$

I think due to the $\cos$, any $x$ going into it will only come out $\in [-1,1]$, then it would have to $= 0$. Just looking for some confirmation.

Thanks in advance.

Answer 1

Recall that for all $x$, $|\cos(x)|\le 1$. Therefore, we can write

$$\left|\frac{\cos(n\pi x)}{n}\right|\le \frac1n <\epsilon$$

whenever $n>\frac1\epsilon$.

Answer 2

"uniform convergence of a sequence of functions" and "uniform continuity of a function" are two different concepts.

One can ask

• (1) if a sequence $f_n$ converges uniformly to some function on the interval $[0,1]$
• (2) or if each $f_n$ is uniformly continuous on the interval $[0,1]$.

(2) has nothing to do with the proof of (1). On the other hand, if you are asking (2) per se, then you might want to know a theorem in real analysis which says that any continuous real function on a closed interval is uniformly continuous.

For (1), all you need is that the cosine function is bounded.

Answer 3

I'm not sure you're clear on the definition of uniform convergence. Specifically, you should be asking if there is a large $N \in \mathbb{N}$ such that for all small $\epsilon > 0$ where $\epsilon$ does not depend on $N$, $n$, or $x$, for all $n>N$, you have the bound

$$\left|f_n(x) - f(x)\right| < \epsilon$$

for some function $f(x)$. In your case, as you said $n$ is going to be dominating the numerator, so the function $f(x)$ that you want to converge to will be $f(x) \equiv 0$.

So, given $\epsilon>0$, can you find the value $N$ such that $\left|f_n(x)\right| < \epsilon$ is true?