**Given:**

The sequence is
an=n5n!
.

**Calculation:**

Plot the points
(n,an), for n=1,2,...,15
on the graph as shown below in Figure 2.

From the graph, it is observed that the smallest value of *N* that corresponds to
ε=0.1
is 9.

Since
959!=0.1627>0.1
,
n5n!<0.1 if n≥10

Plot the points
(n,an), for n=1,2,...,15
on the graph as shown below in Figure 3.

From the graph, it is observed that the smallest value of *N* that corresponds to
ε=0.001
is 12.

Since
11511!=0.0040>0.001
,
n5n!<0.001 if n≥12
.

Therefore, the smallest value of *N* that corresponds to
ε=0.1
is 9 and
ε=0.001
is 12.