Sequence, convergence determine a variable

Question

Determine X, soo the sequence below is convergent:

$$a_n = \frac{n+(2X-4)(-n)^3}{n+1}$$

I'm not sure how to solve this. Should I try to divide them all by $n^3$ to shorten the expression?

Answer 1

You are right. Diving by $n^{3}$ leaves you with $1+\lim \frac {4-2X} {(n+1)n^{-3}}$. so the limit exists only if $X=2$. The limit is $1$ in this case.

[ Separate $\frac n {n+1}$ before dividing by $n^{3}$].