Suppose a new algorithm is designed to predict the growth of the company in future and the error in estimation by the new algorithm is given by an−bnan​−bn​, where anan​ and bnbn​ are the same as defined earlier. Choose the set of correct options.

The error in estimation using the new algorithm is less than the error in estimation using Algorithm 1.

The error in estimation using Algorithm 2 is less than the error in estimation using the new algorithm.

The error in estimation using the new algorithm is less than the error in estimation using Algorithm 3.

The error in estimation using the new algorithm cannot be compared with the error in estimation using Algorithm

what is the correct answer

Let's analyze the new algorithm's error, which is given by:

\[ d_n = a_n - b_n \]

Given:
\[ a_n = \frac{n^2 + 5n}{6n^2 + 1} \]
\[ b_n = \frac{1}{8} + (-1)^n \frac{1}{n} \]

As \( n \to \infty \):
\[ a_n \approx \frac{1}{6} \approx 0.166 \]
\[ b_n \approx \frac{1}{8} = 0.125 \]

So:
\[ d_n \approx a_n - b_n \approx 0.166 - 0.125 = 0.041 \]

Now we evaluate the given statements:

1. **The error in estimation using the new algorithm is less than the error in estimation using Algorithm 1.**

\[ d_n \approx 0.041 \]
\[ a_n \approx 0.166 \]

Since \( 0.041 < 0.166 \), this statement is true.

2. **The error in estimation using Algorithm 2 is less than the error in estimation using the new algorithm.**

\[ b_n \approx 0.125 \]
\[ d_n \approx 0.041 \]

Since \( 0.125 > 0.041 \), this statement is false.

3. **The error in estimation using the new algorithm is less than the error in estimation using Algorithm 3.**

As \( n \to \infty \), \( c_n \to 0 \).

Since \( d_n \approx 0.041 \) and \( c_n \to 0 \), this statement is false.

4. **The error in estimation using the new algorithm cannot be compared with the error in estimation using Algorithm 3.**

The errors can be compared since we have their approximate values as \( n \to \infty \). This statement is false.

Thus, the correct set of options is:

- The error in estimation using the new algorithm is less than the error in estimation using Algorithm 1.

Do you want details or have any questions? Here are some other questions you might be interested in:

1. How to analyze error bounds in multiple algorithms?
2. What factors affect the accuracy of growth prediction algorithms?
3. How to determine the leading error term in asymptotic analysis?
4. What is the impact of oscillating terms on error analysis?
5. How to compare the performance of algorithms with different error rates?

Tip: When comparing errors of different algorithms, it's crucial to analyze their asymptotic behavior and consider the leading terms.

Explore how error estimation works in a new growth prediction algorithm compared to established methods. Analyze the asymptotic behavior and detailed calculations to understand the accuracy levels. Suitable for advanced learners and researchers.

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