Question

How can you derive e( Euler's constant)

Answer 1

Euler's constant, denoted as e, can be derived in several ways. One common method is through the limit definition of the exponential function.

The limit definition of the exponential function states that:

e = lim (n→∞) (1 + 1/n)^n

This means that as the value of n approaches infinity, the expression (1 + 1/n)^n approaches Euler's constant, e.

Let's go through the derivation step by step:

1. Start with the expression (1 + 1/n)^n.

2. Expand the expression using the binomial theorem:

(1 + 1/n)^n = 1 + n*(1/n) + (n*(n-1))/(2!)*(1/n)^2 + (n*(n-1)*(n-2))/(3!)*(1/n)^3 + ...

Note: The binomial theorem expansion can be used because the exponent, n, is a positive integer.

3. Next, simplify the expression:

(1 + 1/n)^n = 1 + 1 + (n-1)/(2!)*(1/n) + (n-1)*(n-2))/(3!)*(1/n)^2 + ...

4. As n approaches infinity, each term involving (1/n) approaches zero, except for the first two terms:

(1 + 1/n)^n = 1 + 1 + 0 + 0 + ...

5. Rearrange the simplified expression:

(1 + 1/n)^n = 2

6. As n approaches infinity, the expression converges to a specific value:

e = lim (n→∞) (1 + 1/n)^n = 2

Therefore, Euler's constant, e, is equal to 2.

Note: This is just one method of deriving e, but it is a commonly used approach. There are other mathematical definitions and methods that can also be used to derive e.