Semiclassical has told me to post an answer to my own question using the technique i have explained to him, now for me to go any further i should state that the original function i am dealing with is:
$
{}_2F_1(\frac{1}{2}, \frac{1-n}{2}; \frac{3}{2};\cos^2 (x))
$
therefore using the power ser...
18
I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can be evaluated by using the Fourier series expansion of $x^2$, where $x\in[-\pi, \pi]$.
$$x^2=\frac...
I am wondering what type of series this this, where you have some constant (let's say 4) to the power of n, summed up where each new exponent keeps going n-1, n-2, n-3, n-4 ... and so on. So,
4^(n) + 4^(n-1) + 4^(n-2) + 4^(n-3) ...
I thought it looked like a geometric series, but all I know abo...
Evaluate $ \sigma(210)$ and $d(63)$
I'm not sure if I got this correct, so here is my attempt.
By Theorem 6.3, suppose we have $n=p_1^{\alpha 1}...p_s^{\alpha s}$, then $d(n) =(\alpha_1 +1)(\alpha_2 +1)...(\alpha_s +1)$ and $\sigma(n) =\frac{p_1^{\alpha_1+1}-1}{p_1-1}\frac{p_2^{\alpha_2+1}-1}...
12
When I showed to my brother how I proved
\begin{equation}
\int_{0}^{\!\Large \frac{\pi}{2}} \ln \left(x^{2} + \ln^2\cos x\right) \, \mathrm{d}x=\pi\ln\ln2
\end{equation}
using the following theorem by Mr. Olivier Oloa
\begin{equation}{\large\int_{0}^{\!\Large \frac{\pi}{2}}} \frac{\cos \left(\! s...
