Problem 29 - Calculus: Early Transcendentals

Question

Determine whether the sequence converges or diverges. If it converges, find the limit. \( a_n = e^{-1/ \sqrt n} \)

Step-by-Step Solution

Answer

The sequence converges, and the limit is 1.

Step 1: Identify the Type of Sequence

The sequence given is \( a_n = e^{-1/ \sqrt{n}} \). It is an exponential sequence where the exponent is \(-1/ \sqrt{n}\). As \(n\) approaches infinity, the term \(\sqrt{n}\) also increases.

Step 2: Consider the Limit of the Exponent

Let's focus on the exponent \(-1/ \sqrt{n}\). As \(n \to \infty\), \(\sqrt{n}\) also approaches infinity, which means \(1/ \sqrt{n}\) approaches 0. Consequently, the whole exponent \(-1/ \sqrt{n}\) approaches 0.

Step 3: Determine the Limit of the Sequence

Since \(-1/ \sqrt{n} \to 0\) as \(n \to \infty\), we look at the limit of the exponential function. For \(a_n = e^{x}\) where \(x \to 0\), \(e^x \to e^0 = 1\). Thus, the sequence \(a_n = e^{-1/ \sqrt{n}}\) converges to 1 as \(n\) goes to infinity.

Step 4: Conclusion on Convergence

Therefore, the sequence \( a_n = e^{-1/ \sqrt{n}} \) converges, and its limit is 1.

Key Concepts

Exponential Sequence

An exponential sequence is a sequence where the terms involve an exponent that is a function of the index, like in the given example, \( a_n = e^{-1/\sqrt{n}} \). Here, the base is the mathematical constant \( e \), commonly known as Euler's number, which is approximately equal to 2.718. In an exponential sequence, the changes in the index \( n \) result in a significant change in the values of the terms due to the exponentiation.

• The base \( e \) is a constant known to be important in many areas of mathematics such as calculus and complex numbers.
• The behavior of the sequence is greatly impacted by the exponent, which can cause rapid growth or decay depending on its value.

Understanding the nature of the exponent in our sequence \(-1/\sqrt{n}\) is crucial to determining its behavior as \( n \) increases.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of a sequence approach as the sequence progresses toward infinity. For the sequence \( a_n = e^{-1/\sqrt{n}} \), recognizing the behavior of the expression \(-1/\sqrt{n}\) is key in finding its limit.

• As \( n \) becomes very large, \( \sqrt{n} \) also grows larger, causing \( 1/\sqrt{n} \) to shrink towards zero.
• Thus, the exponent \(-1/\sqrt{n}\) approaches zero, leading to a crucial evaluation point of the sequence.

Given this, the limit of the sequence involves evaluating the exponential function \( e^x \) at the point where \( x \rightarrow 0 \). With the property that \( e^0 = 1 \), this sequence converges to a limit of 1.

Convergent Sequence

A convergent sequence is defined as a sequence where the terms approach a single finite value as \( n \) approaches infinity. This value is known as the sequence's limit. In our example of the sequence \( a_n = e^{-1/\sqrt{n}} \), we can see that it is convergent.

• This conclusion stems from analyzing the behavior of the sequence's exponent: \(-1/\sqrt{n} \) tends towards zero as \( n \to \infty \).
• Subsequently, \( e^{-1/\sqrt{n}} \) approaches \( e^0 = 1 \).

Therefore, regardless of how large \( n \) becomes, the terms of the sequence will very closely approximate the value 1, reflecting its convergent nature.

Exponential Function Behavior

The behavior of exponential functions like \( e^x \) is characterized by their exponential growth or decay based on the value of the exponent \( x \). This is a crucial observation when examining sequences such as \( a_n = e^{-1/\sqrt{n}} \).

• Exponential growth occurs when the exponent is positive, resulting in rapidly increasing values.
• Conversely, an exponential function decays when the exponent is negative, leading to values that decrease towards zero.
• In our case, as \( -1/\sqrt{n} \) approaches zero, \( e^{-1/\sqrt{n}} \) approaches \( e^0 = 1 \), illustrating a stabilizing behavior because the exponent itself is approaching zero.

Understanding this behavior helps to predict the ultimate behavior of the sequence as \( n \) becomes very large, showing how theoretical concepts of exponential functions translate into tangible outcomes within sequences.