## Problem Statement

Calculate the limit of the numerical sequence:

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

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

## Problem Kuznetsov Limits 6-6