Bashyboy said:Homework Statement
The sequence is [itex]a_n = \frac{n^P}{e^n}[/itex]
Homework Equations
The Attempt at a Solution
If I did L'Hopital's rule P times, would the final product look like:
[itex]P!\times lim_{n \rightarrow \infty} = \frac{1}{e^n}[/itex] ?
A factorial is a mathematical operation denoted by the symbol "!". It is used to represent the product of all positive integers less than or equal to a given number. For example, 5! (read as "five factorial") is equal to 5 x 4 x 3 x 2 x 1, which is equal to 120.
A sequence is a list of numbers that follow a specific pattern or rule. In mathematics, a sequence is represented by terms, which are the individual numbers in the list. For example, the sequence 2, 4, 6, 8, 10... follows the pattern of adding 2 to the previous term to get the next term.
Writing the limit of a sequence with a factorial allows us to find the ultimate value or behavior of the sequence as its terms approach infinity. It helps us understand the long-term trend of the sequence and make predictions about its future values. Additionally, it is a useful tool in many mathematical and scientific fields, such as calculus and statistics.
The symbol for the limit of a sequence is "lim". It is typically written as lim(n → ∞) or limn→∞, where n represents the term number in the sequence and ∞ represents infinity. This notation indicates that we are finding the limit of the sequence as the term number approaches infinity.
To write the limit of a sequence with a factorial, we use the formula lim(n → ∞) a(n)/n!, where a(n) represents the nth term in the sequence. This formula is used when the sequence has a factorial term in its terms. We simply divide the nth term by n! and take the limit as n approaches infinity to find the ultimate value of the sequence.