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?
Consider $\frac{a_1}{a_2 + \cdots + a_{2n}} + \frac{a_3 + \cdots + a_{2n-1}}{a_2 + \cdots + a_{2n}}$. Show that the second term is $\leq 1$. What can you say about the first term?