# Proving the uniform convergence of the Fourier series of $f(x) = x^2$

I calculted the Fourier series for $$f(x)= x^2$$ and I get that it is:

$$\frac{ \pi^{2}}{3} + 4 \sum_{m = 1}^{\infty} \frac{(-1)^m}{m^2} \cos (mx).$$

But the rest of the question asks me to show directly (without using the theorem below) that the series converges uniformly.

Let $$f$$ be a continuous, piecewise differentiable function in $${\rm PC}(2\pi)$$, and suppose that $$f’\in{\rm PC}(2\pi)$$. Then the Fourier series of $$f$$ converges uniformly on $$\Bbb R$$.

Could anyone help me in doing so?

My attempt: I think that the first term is a constant so it will not make a problem for uniform convergence, and the second series is less than or equal $$\sum_{m=1}^{\infty} 1/m^{2}$$ which is a convergent series by $$p$$-test as $$p=2>1$$ . . . but why would this lead to uniform convergence?

• $f(x)=x^2$ on what interval? That's not a periodic function naturally; we have to cut it off somewhere and repeat for a Fourier series to make sense. – jmerry Feb 12 at 5:49
• Try Weierstrass M-test – DragunityMAX Feb 12 at 5:52
• The book did not specify, the author said find the Fourier series of $f(x) = x^2$ only .... so what shall I do in this case? @jmerry – hopefully Feb 12 at 5:52
• I have edited my question @DragunityMAX so I can apply the Weierstrass M-test by the series I mentioned above and it will be the $\sum_{m=1}^{\infty} M_{m}$ in this test? – hopefully Feb 12 at 6:01
• Your choice of a series $\sum_m a_m\cos mx$ implies that it's an even function on an interval of length $2\pi$ - so you chose the interval $[-\pi,\pi]$. That's good - because if we didn't choose a symmetric interval, the periodic function we got by repeating it wouldn't be continuous, and its Fourier series wouldn't converge uniformly. – jmerry Feb 12 at 6:41

$$|\frac{(-1)^m}{m^2} \cos (mx)| \le \frac{1}{m^2}$$ for all $$m \in \mathbb N$$ and all $$x \in \mathbb R.$$
This shows that the series $$\sum_{m = 1}^{\infty} \frac{(-1)^m}{m^2} \cos (mx)$$ converges uniformly on $$\mathbb R.$$