Question

Check whether the following sequences are convergent:

1){4+(-1)^n}

2){(4n+n^2)/(2n^2+3n)}

Answer 1

1) $x_n=4+(-1)^n$

$\lim\limits_{n\to\infty}x_n=\left \{\begin{matrix} 4+(-1)^{n}=5, n=2k \\ 4+(-1)^{n}=3, n=2k-1 \end{matrix}\right.$

because limit  takes two different values ​​then

the sequence is divergent

2) $x_n=\frac{4n+n^2}{2n^2+3n}$

consider the limit

$\lim\limits_{n\to\infty}x_n=\lim\limits_{n\to\infty}\frac{4n+n^2}{2n^2+3n}=\\ =\lim\limits_{n\to\infty}\frac{n^2(\frac{4}{n}+1)}{n^2(2+\frac{3}{n})}=\\ =\lim\limits_{n\to\infty}\frac{\frac{4}{n}+1}{2+\frac{3}{n}}=\frac{1}{2}$

the sequence is convergent