Understanding the Convergence of the Sequence ( a_n sqrt[n]{n} )

The sequence defined by ( a_n sqrt[n]{n} ) is an interesting one to explore in terms of its convergence properties. The goal of this article is to demonstrate, step-by-step, why this sequence converges to 1. We'll explore a few different methods to achieve this, focusing on the AM-GM inequality, and delving into some general principles of limits in calculus.

Proof Using the AM-GM Inequality

One elegant and straightforward way to show the convergence of this sequence is by utilizing the Arithmetic Mean-Geometric Mean (AM-GM) inequality. The AM-GM inequality states that for any non-negative real numbers ( x_1, x_2, ldots, x_n ), the arithmetic mean is always greater than or equal to the geometric mean, and they are equal if and only if ( x_1 x_2 ldots x_n ).

Consider the arithmetic mean of the numbers 1, 1, ..., 1 (n-2 times) and ( sqrt{n} ). Using the AM-GM inequality, we get:

( frac{1 1 cdots 1 sqrt{n}}{n} geq sqrt[n]{1 cdot 1 cdot cdots cdot 1 cdot sqrt{n}} )

Let's break this down. The left side of the inequality is:

[ frac{n-2 sqrt{n}}{n} ]

The right side simplifies to:

[ sqrt[n]{n} a_n ]

Thus, we have:

[ frac{n-2 2sqrt{n}}{n} geq a_n ]

Rearranging this gives:

[ 1 - frac{2}{n} - frac{2}{sqrt{n}} geq a_n ]

As ( n to infty ), the term ( frac{2}{n} ) and ( frac{2}{sqrt{n}} ) both approach 0. Hence, the left side of the inequality converges to 1. Given that ( a_n sqrt[n]{n} geq 1 ) for every integer ( n geq 1 ) (since ( n geq 1 )), and using the squeeze theorem, we conclude that:

[ lim_{n to infty} a_n 1 ]

Alternative Proof Using Exponential Form

Another insightful way to show the convergence of the sequence ( a_n sqrt[n]{n} ) is by transforming the expression using the properties of exponents and logarithms. This method involves rewriting the sequence in a form that is easier to analyze:

[ a_n sqrt[n]{n} e^{frac{ln n}{n}} ]

Now, we need to find the limit of the exponent as ( n to infty ):

[ lim_{n to infty} frac{ln n}{n} 0 ]

This can be shown using L'H?pital's Rule or by recognizing that the logarithmic function grows much slower than the linear function. Hence:

[ lim_{n to infty} e^{frac{ln n}{n}} e^0 1 ]

This confirms that the sequence ( a_n ) converges to 1 as ( n ) approaches infinity.

Conclusion

Both the AM-GM inequality and the exponential transformation provide clear and convincing proofs that the sequence ( a_n sqrt[n]{n} ) converges to 1. These methods not only demonstrate the mathematical rigor but also illustrate the power of inequalities and exponential functions in analyzing limit behaviors.