Let ${x_n}$ and ${y_n}$ be two sequences of positive real numbers. show that $\lim\frac{x_1+x_2+\cdots + x_n}{y_1+y_2+\cdots + y_n}$ = L

Asked Jun 29 '21 at 03:31

Active Jun 29 '21 at 03:31

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Let ${x_n}$ and ${y_n}$ be two sequences of positive real numbers. If $\lim y_n = a$, $a\neq 0$ and $\lim\frac{ x_n}{y_n}$ = L, show that $\lim\frac{x_1+x_2+\cdots + x_n}{y_1+y_2+\cdots + y_n}$ = L

i tried a bunch of things, but none of them worked. i got that $\lim x_n = \frac{L}{a}$, and also tried to write the $\lim\frac{x_1+x_2+\cdots + x_n}{y_1+y_2+\cdots + y_n}$ in the expanded form, also thought that i could use series somehow, but i'm stuck.

Any hints?

asked Jun 29 '21 at 03:31

Vityôk

Here's a direct brute force way. Suppose that for $ n > N$, $ |y_n - a | < \epsilon, |x_n - L/a | < \epsilon$. If we let $ X = \sum_{i=1}^{n-1} x_i, Y = \sum_{i=1}^{n-1} y_i$, then can you show that $ (X + (n-N)( L/a \pm \epsilon) ) / (Y \pm (n-N) ( a \pm \epsilon) $ tends to $ L $ ? – Calvin Lin Jun 29 '21 at 03:41
Use the Stolz–Cesàro theorem – Martin R Jun 29 '21 at 03:52

Let ${x_n}$ and ${y_n}$ be two sequences of positive real numbers. show that $\lim\frac{x_1+x_2+\cdots + x_n}{y_1+y_2+\cdots + y_n}$ = L

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