4 Why is $(2+\sqrt{3})^n+(2-\sqrt{3})^n$ an integer? 3 Prove inequality $\arccos \left( \frac{\sin 1-\sin x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$ 3 $\int_{0}^\pi \frac{\sin(nx)}{\sin x} dx$ 2 How to solve $n^2+1 \mid 2^n+1$ for positive integer $n$? 2 A tough integral and its generalization:

Reputation (382)

 +10 How can I show whether the series $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n(2+(-1)^n)}$ converges or diverges? +10 Why is $(2+\sqrt{3})^n+(2-\sqrt{3})^n$ an integer? +5 General method for solving linear recurrence relations of n terms -2 Which conjecture has remained unsolved the longest?

Questions (7)

 4 General method for solving linear recurrence relations of n terms 3 Closed-form solution to a recurrence of all previous terms 3 Simple way to evaluate $f(n)= \sum_{r=0}^{\infty}r^n x^r$ 1 Which conjecture has remained unsolved the longest? 1 Solution of recurrences containing powers of terms

Tags (31)

 7 calculus × 7 3 contour-integration 7 elementary-number-theory × 3 3 means 4 algebra-precalculus 3 inequality 3 trigonometry 2 diophantine-equations 3 complex-analysis 2 number-theory