1. What is the Mathematical Constant e?

The mathematical constant e is the base of the natural logarithm. It is approximately equal to 2.71828 and is one of the most important numbers in mathematics, appearing in many areas including calculus, complex numbers, and probability.

2. How Does the Calculator Work?

The calculator uses the infinite series expansion:

Where:

• \( n! \) — Factorial of n (product of all positive integers ≤ n)
• The more terms we include, the more accurate the approximation

Explanation: This series converges to e as the number of terms approaches infinity. Each additional term makes the approximation more precise.

3. Importance of e in Mathematics

Details: The number e is fundamental in mathematics because it appears in solutions to differential equations, compound interest calculations, probability theory (Poisson distribution), and the normal distribution.

4. Using the Calculator

Tips: Enter the number of terms you want to include in the series calculation. More terms will give a more accurate result but will require more computation.

5. Frequently Asked Questions (FAQ)

Q1: Why does this series converge to e?
A: This is derived from the Taylor series expansion of the exponential function e^x evaluated at x=1.

Q2: How many terms are needed for good accuracy?
A: About 10 terms give accuracy to 6 decimal places, 20 terms to 15 decimal places.

Q3: Are there other ways to calculate e?
A: Yes, including limits (e = lim (1 + 1/n)^n as n→∞) and continued fractions.

Q4: What is the factorial of 0?
A: 0! is defined as 1, which is why the first term in the series is 1/0! = 1.

Q5: Why is e important in calculus?
A: The function e^x is unique in being equal to its own derivative, making it fundamental in solving differential equations.