Question

Evaluate the limit of the following sequences or state that the limit does not exist. $$a_{n}=\cos \left(0.99^{n}\right)+\frac{7^{n}+9^{n}}{63^{n}}$$

Short answer

In summary, as \(n\) approaches infinity, the limit of the sequence \(a_n = \cos\left(0.99^n\right) + \frac{7^n + 9^n}{63^n}\) is equal to \(1\).

Step-by-step solution

Analyze the components of the sequence individually

We have two main components of the given sequence: 1. \(\cos\left(0.99^n\right)\): The value inside the cosine converges to \(0\) as \(n\) approaches infinity (\(\lim_{n\to\infty} 0.99^n = 0\)). Thus, the cosine function converges to \(\cos(0) = 1\) as \(n\) approaches infinity. 2. \(\frac{7^n + 9^n}{63^n}\): We will simplify this component by factoring out a common term.

Simplify the second component of the sequence

To simplify the second component, we can factor out a \(7^n\) term from the numerator: $$\frac{7^n (1 + 9^n/7^n)}{63^n} = \frac{7^n (1 + (9/7)^n)}{63^n}$$ Now, we can divide the numerator and denominator by a common term \(7^n\), which results in: $$\frac{1 + (9/7)^n}{(9)^n}$$

Evaluate the limit of the second component

Now let's evaluate the limit of the simplified expression \(\frac{1 + (9/7)^n}{(9)^n}\) as \(n\) approaches infinity: $$\lim_{n\to\infty} \frac{1 + (9/7)^n}{(9)^n}$$ As \(n\) approaches infinity, the term \((9/7)^n\) goes to infinity, and all other terms remain constant. Therefore, the fraction converges to \(0\).

Calculate the overall limit of the sequence

Now that we have calculated the limit of each component, we can evaluate the overall limit of the given sequence: $$\lim_{n\to\infty} (\cos\left(0.99^n\right) + \frac{7^n + 9^n}{63^n}) = \lim_{n\to\infty} \cos\left(0.99^n\right) + \lim_{n\to\infty} \frac{1 + (9/7)^n}{(9)^n}$$ $$= 1 + 0$$

State the limit of the sequence

The limit of the given sequence as \(n\) approaches infinity is: $$a_{n}=\lim_{n\to\infty} (\cos \left(0.99^{n}\right)+\frac{7^{n}+9^{n}}{63^{n}}) = 1$$

Key concepts

Convergence of Sequences

Understanding the convergence of sequences is crucial for analyzing how functions behave as they progress towards infinity. A sequence is a list of numbers, usually following a specific formula, where the n-th term is denoted by a_n. A sequence converges to a limit L if the terms become arbitrarily close to L as n increases. Formally, for any small positive number ε, there exists a corresponding integer N such that for all n > N, the difference between a_n and L is less than ε.

If a sequence does not get closer to a single value, it is said to diverge. In the given exercise, we need to find if the sequence a_n = cos(0.99ⁿ) + (7ⁿ + 9ⁿ)/63ⁿ converges or diverges. Evaluating each component of the sequence and following the steps for limits, we can determine the behavior of the entire sequence as n approaches infinity.

Cosine Function Limits

The cosine function is periodic and bounded, meaning its values oscillate between -1 and 1. As a result, limits involving the cosine function depend on the input approaching a particular value. In the context of sequences, if the term inside the cosine function approaches a constant as n becomes large, then cos(a_n) approaches cos(L), where L is the limit of a_n.

In our exercise, the input to the cosine function is 0.99ⁿ, which approaches 0 as n approaches infinity. Hence, we use the limit property of the cosine function to deduce that cos(0.99ⁿ) converges to cos(0) = 1. This result is vitally important as it shows how functions, such as the cosine, can stabilize to a value when the variable within conforms to a specific behavior.

Exponential Expressions

Exponential expressions like 7ⁿ and 63ⁿ are important to understand when looking at sequences. In simple terms, an exponential expression involves a base raised to a power, reflecting repeated multiplication. These expressions can grow very quickly or decay towards zero depending on the base. For bases greater than one, the expression grows; for bases between zero and one, it decays.

In the sequence problem we're discussing, 7ⁿ grows much slower than 63ⁿ, as 63 is 9 times 7. This disparity in growth rates is utilized to simplify the expression and determine the limit. By factoring out common terms and reducing the expression, we can see that the exponential term with a base smaller than one, (9/7)ⁿ, will dominate and push the expression towards zero as n goes to infinity. These steps showcase the fascinating nature of exponential expressions and their impact on the behavior of sequences.