limit of quotient of two series Let $Y_{n} > 0$ for all $ n\in \mathbb{N} $, with $\sum{Y_{n}}= +\infty$.
If $\displaystyle\lim\limits_{n\rightarrow \infty}\frac{X_{n}}{Y_{n}}= a$ then $\displaystyle\lim\limits_{n\rightarrow\infty}\frac{X_{1}+X_{2}+X_{3}+\dots+X_{n}}{Y_{1}+Y_{2}+Y_{3}+\dots+Y_{n}}= a$
I need a idea for solution.
can i supose that $\lim\limits_{n\rightarrow\infty}Y_{n}$ exist ?

$\newcommand{\limti}[1]{\lim\limits_{#1\to\infty}}$This observation is perhaps trivial, but I think it's useful to mention that this is just a different form of Stolz-Cesaro theorem.
Let us have a look at these two claims.

Claim 1: Let $(x_n)$ and $(y_n)$ be real sequences. Let $(y_n)$ be positive, strictly increasing and $\limti n y_n=+\infty$.
  If there exists a limit
  $$\limti n\frac{x_{n+1}-x_n}{y_{n+1}-y_n}=L,$$
  then
  $$\limti n\frac{x_n}{y_n}=L.$$
Claim 2: Let $(a_n)$ and $(b_n)$ be real sequences such that $b_n>0$ for each $n\in\mathbb N$ and
  $\sum b_n=+\infty$. If there exists a limit
  $$\limti n\frac{a_n}{b_n}=L,$$
  then
  $$\limti n\frac{\sum_{k=1}^n a_k}{\sum_{k=1}^n b_k}=L.$$

By taking $a_n=x_{n+1}-x_n$ and $b_n=y_{n+1}-y_n$ we get Claim 2 from Claim 1. Conversely, by taking the partial sums $x_n=\sum_{k=1}^n a_n$ and $y_n=\sum_{k=1}^n b_n$ we get Claim 1 from Claim 2.

Basically with the same proof a slightly more general result can be shown.
The proof can be found e.g. here.

Theorem
  Let $(x_n)$ and $(y_n)$ be real sequences. Let $(y_n)$ be positive, strictly increasing and $\limti n y_n=+\infty$.
  $$
\liminf\frac{x_{n+1}-x_n}{y_{n+1}-y_n}\le
\liminf\frac{x_n}{y_n}\le \limsup\frac{x_n}{y_n}\le
\limsup\frac{x_{n+1}-x_n}{y_{n+1}-y_n}.
$$

