Proving the Limit of ( frac{a^n}{n!} ) as ( n ) Approaches Infinity

In this article, we explore the convergence of the sequence ( x_n frac{a^n}{n!} ) as ( n ) approaches infinity. We will demonstrate that the sequence converges to 0 for any real number ( a ), using both the ratio test for sequences and an in-depth analysis of the factorial function.

The Ratio Test and Convergence

According to the ratio test for sequence convergence, a sequence ( x_n, x_{n1}, x_{n2}, ldots ) converges to 0 if the ratio between consecutive terms converges to a value less than one. Let's apply this to the sequence ( x_n frac{a^n}{n!} ).

First, we find the ratio:

Ratio:

( lim_{n to infty} frac{x_{n 1}}{x_n} lim_{n to infty} frac{frac{a^{n 1}}{(n 1)!}}{frac{a^n}{n!}} lim_{n to infty} frac{a^{n 1} cdot n!}{a^n cdot (n 1)!} lim_{n to infty} frac{a}{n 1} 0 )

Since the limit of the ratio between consecutive terms is 0, which is less than 1, the sequence ( frac{a^n}{n!} ) converges to 0 as ( n ) approaches infinity.

Alternative Proofs and Further Analysis

We will now present an alternative proof and another perspective to reinforce our findings. Given that the limit ( L lim_{n to infty} frac{a^n}{n!} ) is of interest, we can dissect the factorial function ( n! ) and observe its behavior.

Recall that ( n! ) can be expanded as:

Factorial Expansion:

( n! n cdot (n-1) cdot (n-2) cdot ldots cdot 3 cdot 2 cdot 1 )

From this expansion, we can write the sequence as:

Expansion for Sequence Convergence:

( lim_{n to infty} frac{a^n}{n!} lim_{n to infty} frac{a^n}{n^n - a_{n-1} cdot n^{n-1} a_{n-2} cdot n^{n-2} - a_{n-3} cdot n^{n-3} ldots (-1)^{n-1} cdot n!} )

Where the coefficients ( a_{n-r} ) are given by:

Coefficients:

( a_{n-1} sum_{r1}^{n-1} r )

( a_{n-2} sum_{r_1 r_2 1 text{ and } r_1 eq r_2}^{n-1} r_1 r_2 )

( a_{n-3} sum_{r_1 r_2 r_3 1 text{ and } r_1 eq r_2 eq r_3}^{n-1} r_1 r_2 r_3 )

And so on.

Substituting the expanded form of ( n! ) into the limit, we have:

Final Limit Expression:

( L lim_{n to infty} frac{a^n}{n^n - a_{n-1} cdot n^{n-1} a_{n-2} cdot n^{n-2} - a_{n-3} cdot n^{n-3} ldots (-1)^{n-1} cdot n!} )

Each term in the numerator and the denominator can be analyzed as ( n ) approaches infinity:

( lim_{n to infty} frac{a}{n} 0 )

Hence, the entire fraction approaches 0, confirming that the limit is 0 for any real number ( a ).

Moreover, the coefficients ( a_{n-r} ) grow faster than the corresponding powers of ( n ), ensuring that the contributions of higher terms in the factorial expansion become negligible as ( n ) increases. This further justifies the limit behavior.

Conclusion

In conclusion, the sequence ( frac{a^n}{n!} ) converges to 0 as ( n ) approaches infinity, for any real number ( a ). This result is supported by both the ratio test and detailed analysis of the factorial function. Understanding these concepts is crucial for advanced calculus, probability theory, and discrete mathematics.

Keywords

Limit of ( frac{a^n}{n!} ), Sequence Convergence, Ratio Test, Factorial Function