The harmonic series and the resulting curves

The harmonic series and the resulting curves

Nicholas Kampouras

Mechanical Engineer M.Sc.

-------------------------------------------------------------------------------------------------------------------------------------------------------------------

The Harmonic series is 1+1/2+1/3+1/4+1/5+… (see harmonic series)

The sum of the first n terms is approximately:

 

see harmonic series by Jim Loy

γ is the  Euler – Mascheroni constant (see Euler Mascheroni constant)

Let’s consider the harmonic progression:   (see harmonic progression)

α_n = 1/n     1, 1/2, 1/3, 1/4, 1/5, 1/6,…

α1	α2	α3	α4	α5	…
1	1/2	1/3	1/4	1/5	…

Calling y = α_n = 1/n and x = n we have the function y = 1/x,

y1	y2	y3	y4	y5	…
x1 = 1	x2 = 1/2	x3 = 1/3	x4 = 1/4	x5 = 1/5	…

Thus the terms of the harmonic progression and their order in the sequence can be graphically represented by the graph of the function y = 1/x.

Now let’s consider a new sequence b_n with terms α_n as above, in a way that:

b_n = (α₁ + α₂ +…+ α_n)

b1	b2	b3	b4	b5	…
α1	α1 + α2	α1+ α2+ α3	α1+α2+α3 +α4	α1+α2+α3+α4+α5	…
1	1+1/2	1+1/2+1/3	1+1/2+1/3+1/4	1+1/2+1/3+1/4+1/5	…

Calling b_n = y and n = x we have y = lnx + γ.

And the graph of the y = lnx + γ is: