Calculting Limit using Stolz-Cesaro theorem

Problem The sequence of real numbers $(x_n)_{n \geq 1}$ satiesfies $\lim_{n \rightarrow \infty} (x_{2n} + x_{2n+1} )= 315$ and $\lim_{n \rightarrow \infty} (x_{2n} + x_{2n-1}) = 2003$. Evaluate $\lim_{n \rightarrow \infty} (x_{2n}/x_{2n+1})$.

Solution Set $a_n = x_{2n}$ and $b_n = x_{2n+1}$ and observe that $$\frac{a_{n+1} - a_n}{b_{n+1} - b_n} = \frac{(x_{2n+2}+ x_{2n+1})-(x_{2n+1} + x_{2n})}{(x_{2n+3} + x_{2n+2}) -(x_{2n+2} + x_{2n+1})} \rightarrow \frac{2003 - 315}{315-2003} = -1$$

I have some questions about this problem and solution.

1. I think $(x_{2n} + x_{2n+1})$ and $(x_{2n} + x_{2n-1})$ goes to same limit value because $(2n+1)$ and $(2n-1)$ have nothing different, intuitively. Why do they have different limit?

2. The solution uses Stolz-Cesaro theorem to calculate it. Here I wrote the theorem.

Stolz-Cesaro Let $(a_n)_{n \geq 1}$ and $(b_n)_{n \geq 1}$ be two sequences of real numbers.

(i) Assume that $a_n \rightarrow 0$ and $b_n \rightarrow 0$ as $n \rightarrow \infty$. Suppose, moreover, that $(b_n)_{n \geq 1}$ is strictly decreasing for all sufficiently large $n$ and there exists $$\lim_{n \rightarrow \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = l \in \bar{\mathbb{R}}$$. Then there exists $\lim_{n \rightarrow \infty} a_n /b_n$ and moreover, $\lim_{n \rightarrow \infty} a_n/b_n = l$.

(ii) Assume that $b_n \rightarrow +\infty$ as $n \rightarrow \infty$ and that $(b_n)_{n \geq 1}$ is strictly increasing for all sufficiently large $n$. Suppose that there exists $$\lim_{n \rightarrow \infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} = l \in \bar{\mathbb{R}}$$. Then there exists $\lim_{n \rightarrow \infty} a_n /b_n$ and moreover, $\lim_{n \rightarrow \infty} a_n/b_n = l$.

Now return to the question. The solution does not show that [$b_n$ is strictly decreasing, the limit is $0$ and $a_n \rightarrow 0$] or [$b_n$ is strictly increasing and the limit is $+\infty$]. How to prove that it satisfies the condition of Stolz-Cesaro theorem?

Solution 1:

Well the second version of Cesaro-Stolz requires $b_{n} \to \infty$ and strictly increasing nature of $b_{n}$. By changing sign of $b_{n}$ you can see that theorem holds even when $b_{n}$ is strictly decreasing to $-\infty$. This is the case here for $b_{n} = x_{2n - 1}$.

We have by subtraction $$\lim_{n \to \infty}(x_{2n} + x_{2n + 1}) - (x_{2n} + x_{2n - 1}) = 315 - 2003$$ or $$\lim_{n \to \infty}(b_{n + 1} - b_{n}) = -1888$$ which shows that $b_{n}$ is strictly decreasing to $-\infty$. So the Cesaro Stolz theorem applies.

Regarding your first question: The sequences $x_{2n + 1}$ and $x_{2n - 1}$ tend to the same limit provided the limit exists or is $\pm\infty$. But in case the limit is $\pm\infty$ (it is the case for this question) then we are not sure if they tend to $-\infty$ at the same rate. So when we add the term $x_{2n}$ (which tends to $\infty$) to these sequences it is possible that the resulting sequences have different limits.