## Problem Statement

Calculate the limit of the numerical sequence:

$\lim _{n \rightarrow \infty} \frac{2+4+6+\ldots+2 n}{1+3+5+\ldots+(2 n-1)}$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{2+4+6+\ldots+2 n}{1+3+5+\ldots+(2 n-1)}=\lim _{n \rightarrow \infty} \frac{\left(\frac{(2+2 n) n}{2}\right)}{\left(\frac{(1+(2 n-1)) n}{2}\right)}= \\
& =\lim _{n \rightarrow \infty} \frac{(2+2 n) n}{(1+(2 n-1)) n}=\lim _{n \rightarrow \infty} \frac{2 n+2 n^{2}}{2 n^{2}}= \\
& =\lim _{n \rightarrow \infty}\left(\frac{1}{n}+1\right)=0+1=1
\end{aligned}
$$

## Problem Kuznetsov Limits 6-24