Question

Use the following information to determine the limit of the given sequences. $$ \begin{array}{l} \left\\{a_{n}\right\\}=\left\\{\frac{2^{n}-20}{2^{n}}\right\\} ; \quad \lim _{n \rightarrow \infty} a_{n}=1 \\ \left\\{b_{n}\right\\}=\left\\{\left(1+\frac{2}{n}\right)^{n}\right\\} ; \quad \lim _{n \rightarrow \infty} b_{n}=e^{2} \\ \left\\{c_{n}\right\\}=\\{\sin (3 / n)\\} ; \quad \lim _{n \rightarrow \infty} c_{n}=0 \end{array} $$ $$\left\\{a_{n}\right\\}=\left\\{\sin (3 / n)\left(1+\frac{2}{n}\right)^{n}\right\\}$$

Short answer

The limit of the sequence \(\{a_n\}\) is 0 as \(n \to \infty\).

Step-by-step solution

Understanding the Limit of Sequence a_n

The sequence \(a_n = \frac{2^n - 20}{2^n}\) can be simplified as we analyze the behavior of each component as \(n\) approaches infinity. By dividing the numerator and the denominator of \(a_n\) by \(2^n\), we get: \(a_n = 1 - \frac{20}{2^n}\). As \(n\) goes to infinity, \(\frac{20}{2^n}\) goes to 0 because the exponential growth of \(2^n\) is faster than the constant 20. Therefore, the limit of \(a_n\) as \(n\) approaches infinity is 1.

Understanding the Limit of Sequence b_n

The sequence \(b_n = \left(1 + \frac{2}{n}\right)^n\) resembles the form of the exponential limit \(\left(1 + \frac{x}{n}\right)^n \to e^x\) as \(n \to \infty\). Setting \(x = 2\), it follows that \(b_n\) approaches \(e^2\) when \(n\) becomes very large, based on this exponential growth behavior.

Understanding the Limit of Sequence c_n

The sequence \(c_n = \sin\left(\frac{3}{n}\right)\) as \(n \to \infty\) can be understood using the small angle approximation \(\sin(x) \approx x\) when \(x\) is near zero. Here, \(\frac{3}{n} \to 0\) as \(n\) increases, therefore \(c_n = \sin\left(\frac{3}{n}\right)\) will approach 0 as \(n \to \infty\).

Analyzing the Limit of the Given Sequence a_n

The given sequence to analyze is \(a_n = \sin\left(\frac{3}{n}\right)\left(1 + \frac{2}{n}\right)^n\). We already know from Step 3 that \(\sin\left(\frac{3}{n}\right)\) approaches 0 as \(n\) increases. Similarly, from Step 2, \(\left(1 + \frac{2}{n}\right)^n\) approaches \(e^2\). Hence, when multiplying these terms together: \(a_n = \sin\left(\frac{3}{n}\right) \cdot e^2\), \(a_n\) approaches 0 because \(\sin\left(\frac{3}{n}\right)\) is the dominant factor going to zero.

Key concepts

Limit of a Sequence

When determining the limit of a sequence, our main goal is to understand the behavior of the sequence as the index, usually denoted by \( n \), becomes very large. This involves analyzing how each component of the sequence behaves as \( n \) grows without bound.

Let's look at the sequence \( a_n = \frac{2^n - 20}{2^n} \). By dividing both the numerator and the denominator by \( 2^n \), we simplify the sequence to \( a_n = 1 - \frac{20}{2^n} \). As \( n \) approaches infinity, \( \frac{20}{2^n} \) becomes negligible since \( 2^n \) grows much faster than the constant 20. Therefore, the limit of sequence \( a_n \) is simply 1.

In general, while dealing with limits of sequences like this, we look for "dominant" terms—terms that have the most significant impact on the sequence as \( n \) increases. Typically, if a sequence can be broken down into simpler terms where one approaches zero, it's the dominant term that dictates the sequence's limit.

Exponential Limit

Exponential limits are a fascinating area where sequences can model exponential behavior, particularly when the sequence's form is \( \left(1 + \frac{x}{n}\right)^n \). These types of sequences approximate the number \( e \) (Euler's number) when \( n \) becomes very large.

For instance, consider the sequence \( b_n = \left(1 + \frac{2}{n}\right)^n \). By setting \( x = 2 \), we find that this sequence approaches \( e^2 \) as \( n \to \infty \). This behavior is due to the exponential limit theorem, which states that \( \left(1 + \frac{x}{n}\right)^n \) converges to \( e^x \). This is because as \( n \) enlarges, the small increment \( \frac{2}{n} \) applied \( n \) times effectively builds up an exponential growth factor.

Understanding exponential limits is crucial in calculus, as they frequently crop up in contexts involving continuous growth and decay processes, making them foundational in understanding natural logarithms and exponential functions.

Small Angle Approximation

The small angle approximation is a useful concept in sequences involving trigonometric functions. It's based on the principle that for very small angles, measured in radians, \( \sin(x) \) approximates \( x \).

In the sequence \( c_n = \sin\left(\frac{3}{n}\right) \), as \( n \) becomes very large, \( \frac{3}{n} \) approaches zero, making it a small angle. By applying the small angle approximation, we can conclude that \( \sin\left(\frac{3}{n}\right) \approx \frac{3}{n} \). Since \( \frac{3}{n} \) tends to zero as \( n \) increases, it follows that the sequence \( c_n \) also approaches zero.

This approximation is not only helpful in dealing with limits but also in simplifying complex calculations in physics and engineering, where small angular displacements are often involved.