Problem 13
Question
\(9-32\) n Determine whether the sequence converges or diverges. If it converges, find the limit. $$a_{n}=\tan \left(\frac{2 n \pi}{1+8 n}\right)$$
Step-by-Step Solution
Verified Answer
The sequence converges, and its limit is 1.
1Step 1: Recognize the Limit
We need to evaluate the limit of the sequence as \( n \to \infty \). The sequence given is \( a_n = \tan \left( \frac{2n\pi}{1+8n} \right) \). First, we must analyze the behavior of the argument of the tangent function, \( \frac{2n\pi}{1+8n} \), as \( n \to \infty \).
2Step 2: Simplify the Argument
Examine the expression \( \frac{2n\pi}{1+8n} \). As \( n \) becomes very large, the term \( 1 + 8n \) is dominated by \( 8n \). Hence, the expression simplifies to \( \frac{2n\pi}{8n} = \frac{\pi}{4} \).
3Step 3: Calculate the Limit of the Argument
From the simplification, the limit of the argument of the tangent, \( \frac{2n\pi}{1+8n} \), is \( \frac{\pi}{4} \) as \( n \to \infty \).
4Step 4: Find the Limit of the Sequence
Now, substitute the limit of the argument into the tangent function. We have \( \lim_{n \to \infty} a_n = \tan \left( \lim_{n \to \infty} \frac{2n\pi}{1+8n} \right) = \tan \left( \frac{\pi}{4} \right) = 1 \).
5Step 5: Conclusion about the Sequence
Since the limit exists and is finite (\( \tan(\pi/4) = 1 \)), the sequence converges. The limit of the sequence \( a_n \) is 1.
Key Concepts
Tangent FunctionInfinite LimitsSequence Limit Evaluation
Tangent Function
The tangent function is a trigonometric function that relates the angles of a triangle to its opposite and adjacent sides. Mathematically, it's defined as the ratio \( an(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). This function has distinct properties, especially in terms of periodicity and range. The tangent function is periodic with a period of \( rac{\pi}{2}\), which means it repeats its pattern every \( rac{\pi}{2}\) interval. This periodicity is vital in sequence limit problems, as it helps to determine the pattern or behavior of repeating angles.
In our exercise, we encountered an angle formed by the expression \(\frac{2n\pi}{1+8n}\). The key was understanding that as \(n\) becomes very large, the argument simplifies. This understanding enabled us to evaluate the behavior of the tangent function at this specific angle. Remember, the properties of tangent can affect limits significantly, especially due to its asymptotic behavior, which can lead to undetermined results or necessitate careful reevaluation during limit problems.
In our exercise, we encountered an angle formed by the expression \(\frac{2n\pi}{1+8n}\). The key was understanding that as \(n\) becomes very large, the argument simplifies. This understanding enabled us to evaluate the behavior of the tangent function at this specific angle. Remember, the properties of tangent can affect limits significantly, especially due to its asymptotic behavior, which can lead to undetermined results or necessitate careful reevaluation during limit problems.
Infinite Limits
Infinite limits refer to the behavior of a function as the variable approaches infinity. In our exercise context, as \(n\) increases, we analyze how the sequence behaves, specifically the argument of the tangent function. Typically, handling infinite limits requires simplifying expressions to reveal dominant terms. Like in our problem, where \(\frac{2n\pi}{1+8n}\) simplified to \(\frac{\pi}{4}\) for large values of \(n\). This simplification was crucial for determining the approach towards infinity.
It's important to note that infinite limits often involve identifying and managing terms within expressions that contribute significantly as the variable grows large. Once simplified, evaluating the function at this limit becomes straightforward. Hence, recognizing the limit behavior through steps like factoring, division of leading coefficients, or employing L'Hopital's Rule can be very effective in these scenarios.
It's important to note that infinite limits often involve identifying and managing terms within expressions that contribute significantly as the variable grows large. Once simplified, evaluating the function at this limit becomes straightforward. Hence, recognizing the limit behavior through steps like factoring, division of leading coefficients, or employing L'Hopital's Rule can be very effective in these scenarios.
Sequence Limit Evaluation
When evaluating the limits of sequences, the primary objective is to confirm whether the sequence converges or diverges as \(n\) approaches infinity. In our example, after simplifying the argument \(\frac{2n\pi}{1+8n}\) to \(\frac{\pi}{4}\), we substituted this into the tangent function to identify the sequence's limit. The result was \(\tan(\frac{\pi}{4}) = 1\), showing sequence convergence.
Sequence limit evaluation involves these steps:
Sequence limit evaluation involves these steps:
- Identifying and simplifying the expression within the sequence.
- Finding the limit of this simplified expression as \(n\) goes to infinity.
- Applying this limit to the rest of the function (e.g., the tangent function).
- Determining the consequence in terms of sequence convergence or divergence.
Other exercises in this chapter
Problem 13
\(13-16=\) Approximate the sum of the series correct to four decimal places. $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{(2 n) !} $$
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(a) Approximate \(f\) by a Taylor polynomial with degree \(n\) at the number \(a\) . (b) Use Taylor's Formula to estimate the accuracy of the approximation \(f(
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Determine whether the series is convergent or divergent. If it is convergent, find its sum. $$\sum_{k=1}^{\infty} \frac{k(k+2)}{(k+3)^{2}}$$
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