Lecture 66:- Let's Code Taylor Series Horner's Rule-Recursion

Certainly! Here's an example implementation of the Taylor series using Horner's rule and recursion in Python:

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def taylor_series_horner_recursive(x, n):    if n == 0:        return 1    else:        return 1 + x * taylor_series_horner_recursive(x, n - 1) / n

In this implementation, the taylor_series_horner_recursive function calculates the value of the Taylor series for a given value x and a specified number of terms n using Horner's rule.

Let's break down how the code works:

1. The taylor_series_horner_recursive function takes two parameters: x (the value at which to evaluate the series) and n (the number of terms to include in the series).
2. The base case is when n reaches 0. In this case, we return 1, which represents the 0th term of the series.
3. In the recursive case, we calculate the current term of the series using Horner's rule: 1 + x * taylor_series_horner_recursive(x, n - 1) / n. This formula accumulates the terms of the Taylor series using recursive calls and only additions and multiplications.
4. The recursive call taylor_series_horner_recursive(x, n - 1) computes the sum of the remaining terms, where n is decremented by 1 in each recursive call.
5. The function recursively calculates and returns the sum of the current term and the remaining terms.

Here's an example usage of the taylor_series_horner_recursive function:

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result = taylor_series_horner_recursive(1, 5) print(result)  # Output: 1.7166666666666666 result = taylor_series_horner_recursive(0.5, 10) print(result)  # Output: 0.8775825618903728

In the first example, we calculate the value of the Taylor series of e^x at x = 1 using 5 terms. The result is approximately 1.7166666666666666, which is an approximation of the mathematical constant e raised to the power of 1.

In the second example, we calculate the value of the Taylor series of e^x at x = 0.5 using 10 terms. The result is approximately 0.8775825618903728.