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summation formula

mathmaniac

Active member
Mar 4, 2013
188
sigma(1/n)

Is there a formula for it?
 

kaliprasad

Well-known member
Mar 31, 2013
1,322

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
sigma(1/n)

Is there a formula for it?
Also note that the infinite series is divergent, and so that can not possibly have a closed form.
 

kaliprasad

Well-known member
Mar 31, 2013
1,322
Also note that the infinite series is divergent, and so that can not possibly have a closed form.
The above statement is not quite correct as

sigma n = n(n+1)/2 is divergergent but it has a colsed form
 

Prove It

Well-known member
MHB Math Helper
Jan 26, 2012
1,404
The above statement is not quite correct as

sigma n = n(n+1)/2 is divergergent but it has a colsed form
The FINITE series has a closed form. The INFINITE series does not. There is nothing wrong with what I said.
 

mathmaniac

Active member
Mar 4, 2013
188
Why not a formula f(n) such that f(n)-f(n-1)=1/n
Why isn't it possible?
 

kaliprasad

Well-known member
Mar 31, 2013
1,322
The FINITE series has a closed form. The INFINITE series does not. There is nothing wrong with what I said.
I am sorry about my statement. I I meant closed form for the finite sum and then as n tends to infinite. My due apologies
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Why not a formula f(n) such that f(n)-f(n-1)=1/n
Why isn't it possible?
It is neither algebraically possible to obtain a homogeneous difference equation by symbolic differencing, nor to find an elementary particular solution to attempt the method of undetermined coefficients.

So what we do is write:

\(\displaystyle \sum_{k=1}^n\frac{1}{k}=H_n\)

where $H_n$ is the $n$th Harmonic number - Wikipedia, the free encyclopedia.
 

mathmaniac

Active member
Mar 4, 2013
188
Is it possible to figure out whether an inductive formula exists for sigma something?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Do you find a pattern from which you can infer an induction hypothesis?
 

mathmaniac

Active member
Mar 4, 2013
188
Looking for a pattern is not easy,how do you know when to stop looking and conclude there is no formula?
I think most series including reciprocals have no formulae,but some have and is it possible to check?
 

chisigma

Well-known member
Feb 13, 2012
1,704
In...

http://mathhelpboards.com/discrete-...ation-tutorial-draft-part-i-426.html#post2494

... it has been demonstrated that is...

$\displaystyle \sum_{k=1}^{n} \frac{1}{k} = \phi (n) + \gamma\ (1)$

... where $\phi(*)$ is the digamma function, defined as...

$\displaystyle \phi(x) = \frac{d}{d x} \ln x!\ (2)$

... being...

$\displaystyle x! = \int_{0}^{\infty} t^{x}\ e^{- t}\ dt\ (3)$

Kind regards

$\chi$ $\sigma$