Given:
Consider the given sequence as
an=n2e−n
.
Definition used:
If
an
is a sequence and
limn→∞an
exists, then the sequence
an
is said to be converges; otherwise it diverges.
Laws of limits for sequences used:
If
f(x)
and
g(x)
are two functions, then
limx→a[f(x)g(x)]=limx→af(x)limx→ag(x)
and
limx→ag(x)≠0
.
Calculation:
Obtain the limit of the sequence to investigate whether the sequence converges or diverges.
limn→∞an=limn→∞(n2e−n) =limn→∞(n2en)
Since
∞∞
is in indeterminate form, apply the L’Hospital’s Rule.
limn→∞(n2e−n)=limn→∞(ddt(n2)ddt(en)) =limn→∞(2nen)
Since
∞∞
is in indeterminate form, apply the L’Hospital’s Rule.
limn→∞(n2e−n)=limn→∞(ddt(2n)ddt(en)) =limn→∞(2en)
Apply the laws of limits for sequences and simplify the terms.
limn→∞(n2e−n)=limn→∞(2)limn→∞(en) =2∞ =2⋅0 =0
Since
limn→∞an
exists and by using the definition, it can be concluded that the sequence converges to the limit 0.
Therefore, the sequence converges to the limit 0.