91Ó°ÊÓ

A polynomial term is a mathematical expression involving powers of a variable, usually represented as \( n^k \), where \( k \) is a non-negative integer. In the context of our sequence \( a_n = n^5 e^{-n} \), the polynomial term is \( n^5 \). This means that the term has a degree of 5 and as \( n \) increases, \( n^5 \) becomes very large.

Polynomial terms grow at different rates depending on their degree:- For example, \( n^2 \) grows slower than \( n^3 \).
- Raising \( n \) to the 5th power means it grows faster than lower-degree polynomial terms.

Understanding how polynomial terms behave is important in calculus, especially when they are part of a sequence or function that includes other types of terms, like exponential terms. It helps us predict the overall behavior as \( n \) changes.

Exponential terms involve a base raised to a variable exponent, typically expressed as \( b^n \), where \( b > 0 \) is a constant. Common bases include Euler's number, \( e \), which is about 2.718. In our sequence, we have an exponential term \( e^{-n} \).
When \( n \) is positive, \( e^{-n} \) is actually \( \frac{1}{e^n} \), which means it quickly shrinks towards zero as \( n \) increases. This decay rate is much faster than any polynomial can grow, making exponential terms dominate in determining the limit behavior of sequences and functions.

• Exponential decay describes processes where quantities reduce over time, like radioactive decay or cooling.
• This rapid reduction is why \( e^{-n} \) becomes negligible compared to \( n^5 \) for very large \( n \).

Combining an exponential decay term with a polynomial allows us to study complex behaviors in sequences and understand transitions in nature and mathematics.

In calculus, the derivative is a measure of how a function changes as its input changes. For our sequence, we consider the derivative of the function \( f(x) = x^5 e^{-x} \) to understand the behavior.
To compute its derivative, we use the product rule because \( f(x) \) is a product of two functions, \( x^5 \) and \( e^{-x} \). The product rule states:\[ (uv)' = u'v + uv' \]For \( f(x) \), let:- \( u = x^5 \) with \( u' = 5x^4 \)
- \( v = e^{-x} \) with \( v' = -e^{-x} \).
Using the product rule, the derivative \( f'(x) = 5x^4 e^{-x} - x^5 e^{-x} \). This tells us how \( f(x) \) increases or decreases.

By setting \( f'(x) = 0 \), we find critical points where the function potentially changes direction. Here, we find that at \( x = 5 \), the sequence reaches its maximum.

• For \( x < 5 \), \( f'(x) > 0 \) indicates increasing behavior.
• When \( x > 5 \), \( f'(x) < 0 \) reveals decreasing behavior.

This is why after \( n = 5 \), the sequence is strictly decreasing.