## Problem Statement

Calculate the limit of the numerical sequence:

$$
\lim _{n \rightarrow \infty} \frac{1-2+3-4+\ldots+(2 n-1)-2 n}{\sqrt[3]{n^{3}+2 n+2}}
$$
## Solution

$$
\begin{aligned}
& \lim _{n \rightarrow \infty} \frac{1-2+3-4+\ldots+(2 n-1)-2 n}{\sqrt[3]{n^{3}+2 n+2}}= \\
& =\{1-2=3-4=\ldots=(2 n-1)-2 n=-1\}= \\
& =\lim _{n \rightarrow \infty} \frac{-1 \cdot n}{\sqrt[3]{n^{3}+2 n+2}}=\lim _{n \rightarrow \infty} \frac{\frac{1}{n} \cdot(-1) \cdot n}{\frac{1}{n} \sqrt[3]{n^{3}+2 n+2}}= \\
& =\lim _{n \rightarrow \infty} \frac{-1}{\sqrt[3]{1+\frac{2}{n^{2}}+\frac{2}{n^{3}}}}=\frac{-1}{\sqrt[3]{1+0+0}}=\frac{-1}{1}=-1
\end{aligned}
$$

## Problem Kuznetsov Limits 6-26