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From: MatthewB2 on 18 Dec 2009 06:41
Hi,

Is there a specific name for these summations besides "sums of radicals"?

Is there a general formula for the value of the summation (for n less than infinity)?

If there is no general formula, is there a website with the value of these summations calculated to a certain decimal precision?

Do these summations all diverge?
From: Robert Israel on 18 Dec 2009 17:11
MatthewB2 <amintor1975(a)yahoo.com> writes:

> Hi,
>
> Is there a specific name for these summations besides "sums of radicals"?
>
> Is there a general formula for the value of the summation (for n less than
> infinity)?
>
> If there is no general formula, is there a website with the value of these
> summations calculated to a certain decimal precision?
>
> Do these summations all diverge?

x^(1/n) = exp(ln(x)/n) = 1 + ln(x)/n + O(1/n^2) so the sum diverges except in
the trivial case x = 1.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: Raymond Manzoni on 19 Dec 2009 04:24
MatthewB2 a écrit :
> Hi,
>
> Is there a specific name for these summations besides "sums of radicals"?
>
> Is there a general formula for the value of the summation (for n less than infinity)?
>
> If there is no general formula, is there a website with the value of these summations calculated to a certain decimal precision?
>
> Do these summations all diverge?


Hi,

Using the expansion proposed by Robert Israel and summing we may get
following asymptotic expansion (for x>0) :
S_n(x) ~ H_n^(1)*ln(x)/1! + H_n^(2)*ln(x)^2/2! + ...

where H_n^(m) = sum_{k=1}^n 1/k^m

Since H_n^(1) ~ ln(n) + gamma + O(1/n)
and H_n^(m) ~ zeta(m) + O(1/n) for m integer > 1

(gamma is the Euler constant :
<http://en.wikipedia.org/wiki/Euler–Mascheroni_constant> a better
expansion of H_n is provided there)

your asymptotic expansion may be rewritten :
S_n(x) ~ ln(n)*ln(x) + F(ln(x)) + O(ln(x)^2/n)

with F the function defined by :
F(z)= gamma*z + sum_{k=2}^oo zeta(k)*z^k/k!

(this implies that with the choice x^(1/n)-1-ln(x)/n instead of
x^(1/n)-1 your sum would be convergent!)

We could go further and write :
S_n(x) ~ ln(n)*ln(x) + F(ln(x)) - ln(x)*(ln(x)-1)/(2*n) -
ln(x)*(ln(x)^2-3*ln(x)+1)/(12*n^2) + O(1/n^3)
and so on...

I can't provide a name for this F function now but it is rather regular!


Hoping it helped,
Raymond

From: Raymond Manzoni on 19 Dec 2009 11:58
Raymond Manzoni a écrit :
> F the function defined by :
> F(z)= gamma*z + sum_{k=2}^oo zeta(k)*z^k/k!
>
> I can't provide a name for this F function now but it is rather regular!

I'll just add that the Laplace transform of the F function is

LF(s)= gamma/s^2 + sum_{n=2}^oo zeta(n)/s^(n+1)
LF(s)= -psi(1-1/s)/s^2
= -(ln(Gamma(1-1/s)))'

with psi the digamma function :
<http://en.wikipedia.org/wiki/Digamma_function#Taylor_series>

Fine continuation!
Raymond
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