Whether the sequence converges or diverges and obtain the limit if the sequence converges.

The sequence is converges to the limit 0.

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.