Question

Determine whether the sequence is convergent or divergent. If it is convergent, find its limit\(\left\{ {{{{\rm{( - 10)}}}^{\rm{n}}}{\rm{/n!}}} \right\}\).

Short answer

The given equation is the sequence that converges to \(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \frac{{{\rm{1}}{{\rm{0}}^{\rm{n}}}}}{{{\rm{n!}}}}{\rm{ = 0}}\).

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.

Even though this is a sequence, the convergence rules are slightly different than for Series.

\(\left\{ {\frac{{{{{\rm{( - 10)}}}^{\rm{n}}}}}{{{\rm{n!}}}}} \right\}\)

So, the theorem is

If

\(\begin{array}{l}\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \left| {{{\rm{a}}_{\rm{n}}}} \right|{\rm{ = 0}}\\\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } {{\rm{a}}_{\rm{n}}}{\rm{ = 0}}\end{array}\)

So, without the alternating sign, one can look at the limit

\(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \frac{{{\rm{1}}{{\rm{0}}^{\rm{n}}}}}{{{\rm{n!}}}}{\rm{ = 0}}\)

\(\mathop {{\rm{lim}}}\limits_{{\rm{n}} \to \infty } \frac{{{\rm{1}}{{\rm{0}}^{\rm{n}}}}}{{{\rm{n!}}}}{\rm{ = }}\frac{{ \cdots {\rm{ \times 10 \times 10 \times 10 \times 10}} \cdots \cdots {\rm{10}} \cdots }}{{ \cdots {\rm{6 \times 24 \times 120 \times 720}} \cdots \cdots \cdots {\rm{3628800}} \cdots }}\)

As a result of the theorem above, the original sequence's limit is also 0. The series converges because the limit exists.