Calculator with E

Calculate the approximation of 'e' using the series expansion.

Understanding Euler's Number (e)

Euler's number, denoted by the symbol e, is a fundamental mathematical constant approximately equal to 2.71828. It is one of the most important numbers in mathematics, appearing in various fields including calculus, compound interest, probability, and more. It is an irrational and transcendental number, meaning its decimal representation never ends and it cannot be a root of a non-zero polynomial equation with integer coefficients.

The Mathematical Basis: Series Expansion

One of the most common ways to define and approximate e is through its infinite series expansion:

e = Σ (1/n!) from n=0 to ∞
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + ...

Where n! (n factorial) is the product of all positive integers up to n. By definition, 0! = 1.

This calculator approximates e by summing the first N terms of this series. The more terms you include (a larger value for 'n'), the closer the approximation will be to the true value of e.

How the Calculator Works:

• Number of Terms (n): This input field determines how many terms from the series expansion are used in the calculation. A higher number of terms leads to a more accurate result.
• Factorial Calculation: For each term 1/k!, the factorial of k (where k ranges from 0 up to n-1) is calculated.
• Summation: The reciprocals of these factorials (1/0!, 1/1!, 1/2!, etc.) are summed together.
• Result: The final sum is displayed as the approximation of e.

Use Cases and Significance:

• Continuous Compound Interest: e is the base of the natural logarithm and is crucial in formulas describing continuous growth, such as continuously compounded interest. The formula A = P * e^(rt), where P is principal, r is rate, and t is time, directly uses e.
• Natural Growth and Decay: Many natural processes, like population growth or radioactive decay, can be modeled using exponential functions with base e.
• Probability and Statistics: e appears in the normal distribution (Gaussian) function, which is fundamental in statistics.
• Complex Analysis and Calculus: The exponential function e^x has unique properties in calculus, such as its derivative being itself.
• Educational Tool: This calculator serves as a practical demonstration of how an infinite series can converge to a specific numerical value, illustrating a key concept in calculus and numerical methods.