MathDB

2

Part of 2023 Brazil Undergrad MO

Problems(1)

find the sum of the inverses of the central binomial coefficients

Source: Brazilian Mathematical Olympiad 2023, Level U, Problem 2

10/21/2023
Let an=1(2nn),∀n≤1a_n = \frac{1}{\binom{2n}{n}}, \forall n \leq 1.
a) Show that ∑n=1+∞anxn\sum\limits_{n=1}^{+\infty}a_nx^n converges for all x∈(−4,4)x \in (-4, 4) and that the function f(x)=∑n=1+∞anxnf(x) = \sum\limits_{n=1}^{+\infty}a_nx^n satisfies the differential equation x(x−4)f′(x)+(x+2)f(x)=−xx(x - 4)f'(x) + (x + 2)f(x) = -x.
b) Prove that ∑n=1+∞1(2nn)=13+2π327\sum\limits_{n=1}^{+\infty}\frac{1}{\binom{2n}{n}} = \frac{1}{3} + \frac{2\pi\sqrt{3}}{27}.
functionreal analysisConvergencebinomial coefficients