Question

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit\({{\rm{a}}_{\rm{n}}}{\rm{ = }}\frac{{{{\rm{n}}^{\rm{3}}}}}{{{\rm{1 + }}{{\rm{n}}^{\rm{2}}}}}\).

Short answer

The given equation is the sequence of divergent \(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } {{\rm{a}}_{\rm{n}}}{\rm{ = }}\infty \).

Step-by-step solution

 Cauchy convergence sequence.

A series converges if and only if the sequence of partial sums is a Cauchy sequence, according to the Cauchy convergence criterion. A numerical sequence converges if and only if it is a Cauchy sequence.

If and only if, a sequence is said to converge \(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } {\rm{ }}{{\rm{a}}_{\rm{n}}}\)is a finite constant.

Evaluation.

\(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } {{\rm{a}}_{\rm{n}}}{\rm{ = }}\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \frac{{{{\rm{n}}^{\rm{3}}}}}{{{\rm{1 + }}{{\rm{n}}^{\rm{2}}}}}\)

Divide by \({{\rm{n}}^{\rm{3}}}\),

\(\begin{array}{c}\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } {{\rm{a}}_{\rm{n}}}{\rm{ = }}\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \frac{{\frac{{{{\rm{n}}^{\rm{3}}}}}{{{{\rm{n}}^{\rm{3}}}}}}}{{\frac{{{\rm{1 + }}{{\rm{n}}^{\rm{2}}}}}{{{{\rm{n}}^{\rm{3}}}}}}}\\{\rm{ = }}\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \frac{{\rm{1}}}{{\frac{{\rm{1}}}{{{{\rm{n}}^{\rm{3}}}}}{\rm{ + }}\frac{{{{\rm{n}}^{\rm{2}}}}}{{{{\rm{n}}^{\rm{3}}}}}}}\\{\rm{ = }}\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \frac{{\rm{1}}}{{\frac{{\rm{1}}}{{{{\rm{n}}^{\rm{3}}}}}{\rm{ + }}\frac{{\rm{1}}}{{\rm{n}}}}}\\{\rm{ = }}\frac{{\rm{1}}}{{{\rm{0 + 0}}}}\\\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } {{\rm{a}}_{\rm{n}}}{\rm{ = }}\infty \end{array}\)

So, the given equation is the sequence of divergent \(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } {{\rm{a}}_{\rm{n}}}{\rm{ = }}\infty \).