Exponential Calculation Using Taylor Series

Project Overview

This project implements a Python program to compute the exponential function $e^x$ using its Taylor Series Expansion. The program also incorporates optimizations to handle negative values of $x$, prevent factorial overflow, and determine a stopping criterion for series convergence.

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Objectives

1. Implement Taylor Series: Calculate $e^x$ using the Taylor series formula up to $n$ terms, where $x$ and $n$ are user inputs.
2. Handling Negative Values: Evaluate $e^x$ in two ways for $x < 0$:
   • Directly using the series.
   • Using the transformation $y = -x$ and computing: $e^x = \frac{1}{e^{-x}}$
3. Avoiding Overflow: Optimize factorial calculations to prevent overflow by performing intermediate divisions.
4. Convergence Criterion: Determine the appropriate stopping condition for the Taylor series expansion.
5. Numerical vs Analytical Methods: Analyze the differences and types of errors in the results.
6. Symbolic Computation: Explore symbolic computation to verify results and analyze performance.

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Requirements

• Python 3.x
• Libraries:
  • math (for numerical operations)
  • sympy (for symbolic computation)

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Theoretical Background

1. Taylor Series Expansion

The exponential function $e^x$ can be approximated by the Taylor Series: $e^x = \sum_{k=0}^n \frac{x^k}{k!}$ (where $n$ is the number of terms).

For negative $x$, two approaches are considered:

1. Direct calculation using $x$ in the series.
2. Transforming $x$ to $y = -x$ and using: $e^x = \frac{1}{e^{-x}}$

2. Overflow Prevention

The factorial $k!$ grows rapidly, leading to overflow. To mitigate this, the $k$-th term in the series $\frac{x^k}{k!}$ is computed iteratively:

This avoids explicit computation of $k!$.

3. Stopping Criterion

The series calculation terminates when the relative change between consecutive terms becomes negligibly small: $\frac{\text{term}_k}{\text{sum}} < \text{threshold}$ where threshold is a small predefined value, e.g., $10^{-6}$.

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Numerical vs Analytical Methods

Aspect	Analytical Method	Numerical Method
Definition	Exact solutions derived from formulas.	Approximations using iterative calculations.
Nature	Symbolic and exact.	Approximate and dependent on precision.
Example	$e^x = \lim_{n \to \infty} (1 + x/n)^n$	Taylor series expansion for $e^x$.

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License

This project is licensed under the MIT License.

You can read the full terms of the license in the LICENSE file or visit: