I need to prove that if $(a_n)$ is decreasing and $\displaystyle \sum a_n = + \infty$. Then $\lim {\frac{a_1 + a_3 + \dotsb + a_{2n-1}}{a_2 + a_4 + \dotsb + a_{2n}}} = 1$
I proved that $\displaystyle \lim {\frac{a_1 + a_3 + \dotsb + a_{2n-1}}{a_2 + a_4 + \dotsb + a_{2n}}} \geq 1$. Someone has any idea to finish the demonstration?