# Math Is Fun Forum

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## #1 2016-12-25 23:38:14

Member
From: Planet Mars
Registered: 2016-11-15
Posts: 821

### A direct formula for HP

We know that there are 3 types of progressions Arithmetic, Geometric and harmonic. The Formula for sums of AP and GP are given below:

A.P.→ [n(a1+an)]/2=[n{2a1+(n-1)d}]/2
G.P→ [a1(1-rⁿ)]/(1-r)
where, a1= 1st term
an=last term
d= common difference
r=Common ratio
n=no. of terms                                                                                                                                  n
So can anyone tell  me a similar direct formula for HP(Harmonic progression) also. And yeah I had found out that ∑   1/k ≈In(n) + γ where,γ= Euler Mascheroni const.
k=1                                  ≈ 0.5772156649015.....
Can anyone please prove it.

Practice makes a man perfect.
There is no substitute to hard work
All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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## #2 2016-12-25 23:40:59

Member
From: Planet Mars
Registered: 2016-11-15
Posts: 821

### Re: A direct formula for HP

Note: Harmonic Progression are the reciprocals of Arithmetic progression. It is in the form 1/a+1/b+1/c+1/d..... , where a,b,c,d,.... are in AP.

Practice makes a man perfect.
There is no substitute to hard work
All of us do not have equal talents but everybody has equal oppurtunities to build their talents.-APJ Abdul Kalam

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## #3 2016-12-26 00:35:17

zetafunc
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Registered: 2014-05-21
Posts: 2,436
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### Re: A direct formula for HP

You can prove that using summation by parts. In fact:

For your first question, read this: https://brilliant.org/wiki/harmonic-pro … roximation

Last edited by zetafunc (2016-12-26 00:35:35)

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## #4 2016-12-26 04:53:34

bobbym
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From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

### Re: A direct formula for HP

The Euler Mascheroni constant was my favorite when I was first starting out. Do you know that we do not even know whether or not this number is irrational?!

Hilbert mentioned the irrationality of gamma as an unsolved problem that seems "unapproachable" and in front of which mathematicians stand helpless.

Back in my day it was extremely difficult to calculate to many places, one that I could not do. It occurs in the CCP problem too.

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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## #5 2016-12-26 06:11:07

zetafunc
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Registered: 2014-05-21
Posts: 2,436
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### Re: A direct formula for HP

It also appears in the Dirichlet divisor problem, an unsolved problem about divisors.

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## #6 2016-12-26 16:46:33

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,995
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### Re: A direct formula for HP

The blue area converges to Gamma.

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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