Question

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

Short answer

The limit of the sequence \((1 + \frac{2}{n})^{2n}\) is \(e^{4}\) as \(n\) approaches infinity.

Step-by-step solution

Identify the Sequence

We need to find the limit of the sequence \(\{a_n\} = \{(1 + \frac{2}{n})^{2n}\}\). Observing the form, this resembles expressions related to the number \(e\).

Apply Exponential Limit

Recall that \(\left(1 + \frac{1}{n}\right)^n\) converges to \(e\) as \(n\) approaches infinity. We can use this concept to evaluate our sequence instead by rewriting the term.

Reformulate the Expression

The sequence can be rewritten as \(\left[(1 + \frac{2}{n})^n\right]^2\). This is because \((1 + \frac{2}{n})^{2n} = \left[(1 + \frac{2}{n})^n\right]^{2}\).

Discover Convergence of a Sub-sequence

Recognize that \((1 + \frac{2}{n})^n\) is similar to \((1 + \frac{1}{k})^k\) with \(k \to \infty\), where it converges to \(e^{2}\). Thus, \((1 + \frac{2}{n})^n\) converges to \(e^{2}\).

Calculate the Limit

Since \((1 + \frac{2}{n})^n\) converges to \(e^{2}\), the expression \((1 + \frac{2}{n})^{2n} = [(1 + \frac{2}{n})^n]^2\) will converge to \((e^{2})^2 = e^{4}\).

State the Final Result

Thus, the limit of the sequence \(\{a_n\} = \{(1 + \frac{2}{n})^{2n}\}\) as \(n\) approaches infinity is \(e^{4}\).

Key concepts

Convergence

Convergence is a crucial concept when analyzing the behavior of sequences as the number of terms grows indefinitely. In the context of sequences, convergence refers to the tendency of a sequence to approach a specific value, known as the limit, as the term number, usually denoted as \( n \), becomes very large.

• For example, if we consider the sequence \( \{a_n\} = \{ (1 + \frac{2}{n})^{2n} \} \), its limit can be determined by observing how the sequence behaves as \( n \to \infty \).
• A sequence is said to converge to a limit \( L \) if for every positive number \( \epsilon \), there exists a positive integer \( N \) such that for all \( n > N \), the absolute difference \( |a_n - L| < \epsilon \).

In our case, since exponential expressions often relate to natural exponential growth \( e \), we reformulate our sequence using properties of \( e \) to find it converges to \( e^4 \). Convergence is essential as it helps predict the long-term behavior of functions and systems.

Exponential Functions

Exponential functions often appear in the study of limits and convergence because of their unique growth characteristics that can neatly predict long-term behaviors. An exponential function is typically of the form \( f(x) = a^x \), where \( a \) is a positive constant.

• In sequences, exponential expressions can often be linked through reformulation to the natural number \( e \), which arises from the limit \( \left(1 + \frac{1}{n}\right)^n \) as \( n \to \infty \).
• Our sequence \( \{ (1 + \frac{2}{n})^{2n} \} \) can be expressed as \( [(1+\frac{2}{n})^n]^2 \), which relates to exponential growth.

By expressing a sequence in a format involving exponential functions like \( e \), it becomes easier to apply known limits and properties of \( e \), simplifying the analysis and calculation of limits, leading ultimately to a determination of the sequence’s behavior, in this case, resulting in a limit of \( e^4 \).

Sequence and Series Analysis

The analysis of sequences and series is a fundamental aspect of mathematical analysis, allowing us to predict and understand the behavior of complex mathematical functions over an infinite domain. A sequence is an ordered list of numbers, which can often be expressed in form similar to series and is crucial when employing limits.

• By analyzing the term structure, as in \( \{ (1 + \frac{2}{n})^{2n} \} \), insights can be gained into the sequence's ultimate trends.
• In the case of this sequence, recognizing patterns with known sequences such as that involving \( e \) aids in transforming and simplifying sequence expressions.

Through sequence and series analysis, you can break down complex expressions into manageable parts to quickly identify convergence behavior and limits. This type of analysis is not limited to mere numerical calculation but extends to providing deep insights into asymptotic behaviors which are critical in both theoretical and applied mathematics.