Evaluate Power 5 Using Two Approaches

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Evaluate e power −5 using two approaches: the series expansion of e^−x as 1 − x + x²/2 − x³/3! + ... and the reciprocal of e^x expressed as 1 / (1 + x + x²/2 + x³/6 + ...). Compare these approximate values with the true value of 6.737947×10⁻³. Use 20 terms in each series to evaluate e⁻⁵ and compute the true and approximate relative errors as more terms are included.

Paper For Above instruction

Numerical approximation of exponential functions is fundamental in computational mathematics, particularly when evaluating exponential values for large or complex arguments where direct computation might be impractical or computationally expensive. The goal of this paper is to evaluate e⁻⁵ using two series expansion methods, analyze the accuracy via error analysis, and compare with the known (true) value. Moreover, this exploration underscores the impact of series truncation on the precision of the approximation and highlights the significance of choosing suitable series representations in numerical computations.

In the first approach, the exponential function e^−x is expanded as a power series derived from its Taylor series expansion centered at zero:

e^−x = 1 − x + x²/2 − x³/3! + x⁴/4! − ...

For x = 5, the series becomes an alternating sum of increasingly smaller terms. Truncating this expansion at 20 terms ensures a high degree of accuracy, given the exponential decay of the terms' magnitude. Each added term reduces the approximation error, approaching the true value. To quantify the accuracy, the approximation is compared with the known value of e⁻⁵, approximately 6.737947×10⁻³, and the relative error is calculated as

Relative Error = |Approximate Value − True Value| / |True Value|.

This process demonstrates convergence properties of the Taylor series for the exponential function, illustrating how many terms are required to attain a specified accuracy threshold, especially for larger values of x where convergence is slower.

The second approach involves expressing e^−x as the reciprocal of e^x:

e^−x = 1 / e^x = 1 / (1 + x + x²/2 + x³/6 + ...)

Here, the denominator is expanded as a power series using the binomial or Taylor expansion of e^x. Computing the reciprocal of this series provides an alternative approximation of e^−x. Similar to the first approach, truncating after 20 terms balances accuracy against computational effort, and the resulting value is compared with the true value to evaluate the relative error. The comparison between the two methods reveals their respective convergence rates and numerical stability, especially considering the alternating signs in the series that may lead to cancellation errors.

Both methods exemplify classical techniques in numerical analysis to approximate exponential functions, vital in fields like scientific computing, engineering, and physics. Their efficacy depends on convergence properties and the magnitude of x; as |x| increases, the series convergence slows, demanding more terms for high accuracy.

Furthermore, understanding the magnitude of errors arising from series truncation aids in designing more efficient algorithms that balance precision with computational efficiency. The series expansion method for e^−x illustrated here is a fundamental approach, and the comparison provides insight into good practices for numerical computation of exponentials, especially in environments with limited precision or computational resources.

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