# How do you calculate Euler's Number?

##### 2 Answers

**Note**

This solution provides an estimate for Euler's Number (or

There may be differing opinions, but I would of thought that simplest derivation is to use the Maclaurin Series for

The Maclaurin Series for

# e^x = 1 + x + (x^2)/(2!) + (x^3)/(3!) + (x^4)/(4!) + (x^5)/(5!) + ... #

So the power series sought for

# e = 1 + 1 + (1)/(2!) + (1)/(3!) + (1)/(4!) + (1)/(5!) + ... #

We can compute an approximating fraction by truncating the series as we see fit. For example, If we restrict ourself to the first **five** terms, we get:

# e ~~ 1 + 1 + (1)/(2!) + (1)/(3!) + (1)/(4!)#

# \ \ \ = 1 + 1 + 1/2+1/6+1/24#

# \ \ \ = 65/24 #

# \ \ \ = 2.708333 #

We can compare this to a calculator answer for the Euler Constant,

# e = 2.7182818 ... #

And for further comparison, if we take further terms we get:

# 6 # terms:# e~~65/24+1/120 #

# " " = 163/60 #

# " " = 2.716666 ... #

# 7 # terms:# e~~ 163/60 + 1/720 #

# " " = 1957/720 #

# " " = 2.718055 ... #

# 8 # terms:# e~~ 1957/720 + 1/5040 #

# " " = 685/252#

# " " = 2.718253 ... #

The Euler-Mascheroni constant is defined as:

that is as the difference between the

Since the limit is in indeterminate form we must prove it is convergent. Write

Then:

Now note that using Taylor's formula with Lagrange's rest and

So:

and:

As the series:

is convergent based on the

is convergent and

Incidentally this proves that the partial sums of the harmonic series diverge with the same order as