The Euler-Mascheroni Constant

Arpita Bhattacharya
4 min readDec 24, 2022

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Objective of this article: Defining and calculating this value numerically. The constant’s would appearances in Mathematics.

Assumed knowledge: Integration (Calculus), Concept of convergence

Definition

Known also as the Euler’s constant (not to be confused by e = 2.718281828459045…), the Euler-Mascheroni Constant which is denoted by the lower-case gamma (γ) is defined as,

“The limit as n → ∞ of the difference between the harmonic series and natural log of n.”

Or,

This constant with a value of approximately γ = 0.57721566490153286060651209008240243104215933593992… does exist. But it is still not known if the constant is transcendental or algebraic. In fact it is not even known if γ is either rational or irrational.

If this constant is rational, the denominator must be greater than 10²⁴²⁰⁸⁰.

Calculation of the value of γ

Let’s substitute various values of n in the above defined equation of γ,

We can see here that as n increases and becomes larger and larger, or, as it tends to infinity, the value comes closer to the initially defined value of this constant. This number is the Euler-Mascheroni Constant.

Graphical representation of γ

The orange area under the y = 1/x curve is the natural log function. This is because the integration of, or area under the graph y = 1/x is ln x.

The green area of the bars is graphic representation of the harmonic series.

The difference between these two areas shaded by the blue dots (below) represents the Euler — Mascheroni constant,

Proof of Convergence

We know that as n tends to infinity, the function that defines the Euler — Mascheroni constant converges to a certain constant value. But how do we prove this convergence without any brute force?

So, we know that,

Where,

Let’s take,

which gives the constant when n tends to infinity.

To prove convergence, we’ll

  1. Prove that the sequence is decreasing and then,
  2. Prove that it is bounded.

I. SEQUENCE IS DECREASING

Now, expressing ln⁡(1+x) using the Taylor series,

Taking x = -1/n,

Using (3) in (2),

Thus,

Hence, the sequence is monotonically strictly decreasing.

II. SEQUENCE IS BOUNDED

a) Upper Bound Of Sequence

As we now know that the sequence is strictly decreasing, the first term of the sequence would be the largest term.

So, the first term is,

Hence, the upper bound is 1.

a) Lower Bound Of Sequence

We know that,

This can be written as,

Now,

We can define e^x as

If we omit the case where x = 0,

So,

Taking natural log on both sides,

From (5),

Taking summation on both sides,

From (4),

Adding H_n on both sides of equation,

Or,

Hence, the lower bound is 0.

Thus, G_n is a monotonically strictly decreasing sequence with bounds,

Hence, the sequence converges to a limit γ , such that, 0 < γ < 1.

The limit of the sequence G_n as n → ∞ is the Euler — Mascheroni constant.

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Arpita Bhattacharya
Arpita Bhattacharya

Written by Arpita Bhattacharya

23 | Masters in Math from Lund University, Sweden | Undergraduate from Warwick Uni, UK | STEM Enthusiast |

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