Recent content by δοτ

Absolutely nothing other than simplifying. Think cancel.

Yea, I think that's totally the problem. In the book they kind of glossed over that the negatives disappeared at some point and a coefficient of 2 popped out front. But the negative sums would be the same as the positive sums here because c_n is even. Thanks for guiding me to my error.

I'm positive it's correct. \frac{1}{2\pi} \left(\frac{\pi^3}{3} - \frac{(-\pi)^3}{3}\right) = \frac{1}{2\pi} * \frac{2\pi^3}{3} = \frac{\pi^2}{3}

Homework Statement Let f(x) = x^2 on [-\pi,\pi]. Computer the Fourier Coefficients of the 2π-periodic extension of f. Use Dirichlet's Theorem to determine where the Fourier Series of f converges. Use the previous two conclusions to show that \sum_{n=1}^\infty \frac{1}{n^2} =...

Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...

Yes, the first two terms are the derivative of \displaystyle\frac{e^{3v}}{v}. Rewrite them as such and simply use the Fundamental Theorems of Calculus. It is find the way it is.

In response to your PM, the EVT can be applied to say that, in addition (without loss of generaltiy) c_1 and c_2 are the minimum and maximum on this interval, respectively. We can say this because continuous function must attain their maximum and minimum. Now the IVT can be applied to...

I assume r, b, and h are constants. If so, split the integral into 2\displaystyle\int\limits_{r-b}^r \dfrac{hx}{b} \sqrt{r^2 - x^2} dx + 2\int\limits_{r-b}^r \frac{h(b-r)}{b} \sqrt{r^2 - x^2} dx Then the first can be done with the substitution you've tried already, and the second will...

If we suppose S contains more than one points then are a \le c_1 < c_2 \le b such that f(c_1) \neq f(c_2). Now the EVT can be applied to say something about the relationship of these. Once that's established the MVT will show that it must be an interval. This actually says something quite...

You changed notation in b when the norm of the partial was taken (it should be the partial w.r.t \phi), but other than that it is all correct.

Homework Statement Let f be a continuously differentiable function on the interval [0,2\pi], where f(0) = f(2\pi) and f'(0) = f'(2\pi). For n = 1,2,3,\dotsc, define a_n = \frac{1}{2\pi} \int_0^{2\pi} f(x) \sin(nx) dx. Prove that the series \sum_{n=1}^\infty |a_n|^2 converges...