잡담을 위한 소박한 장소

Problem 1. Prove that[각주:1] \begin{equation*} \sum_{n=1}^{\infty} \frac{1}{n^4 \binom{2n}{n}} = \frac{17\pi^4}{3240}. \tag{1} \end{equation*} Proof. We divide the proof into several steps. 1. Reduction to an integral representation Let $S$ denote the summation in question. By using the Lemma 1 in Today's Calculation 29, we can represent $S$ as an integral. Then by the successive application of i..

Problem 1. Show that[각주:1] \begin{equation*} \int_{0}^{\infty} x \left\{ (2S(x) - 1)^2 + (2C(x) - 1)^2 \right\}^{2} \, \mathrm{d}x = \frac{16 \log 2 - 8}{\pi^2}, \tag{1} \end{equation*} where $S(x)$ and $C(x)$ denote the Fresnel integrals defined by \begin{align*} S(x) = \int_{0}^{x} \sin \left( \tfrac{1}{2} \pi t^2 \right) \, \mathrm{d}t \quad \text{and} \quad C(x) = \int_{0}^{x} \cos \left( \t..

Problem 1. Show that[각주:1] the product \begin{equation*} P = \left( \frac{2}{1} \right)^{\frac{1}{8}} \left( \frac{2 \cdot 2}{1 \cdot 3} \right)^{\frac{3}{16}} \left( \frac{2 \cdot 2 \cdot 2 \cdot 4}{1 \cdot 3 \cdot 3 \cdot 3} \right)^{\frac{6}{32}} \left( \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 4 \cdot 4 \cdot 4 \cdot 4}{1 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 3 \cdot 5} \right)^{\frac{10..