Re: partial sum for...

From: G. A. Edgar (edgar_at_math.ohio-state.edu.invalid)
Date: 08/30/04 

Date: Mon, 30 Aug 2004 09:14:41 -0400

In article <cguiv0$d8h$1@rzcomm2.rz.tu-bs.de>, Peter Unger
<punger@gmx.de> wrote:

> hi,
>
> I'm looking for a partial sum quation for
>
> s_n = \sum_{i=1}^{n} i^\alpha
>
> \alpha should be variable (also < 0)
> is there a euqation f(n) = s_n
>
> regards Peter
>
>

Can be written in terms of the Hurwitz zeta function...

In Maple's notation:
  Zeta(-a) - Zeta(0,-a,m+1) = sum(i^a,i=1..m)

examples (I called the equation above E1):

> subs({a=5,m=11},E1);evalf(%);
                                            11
                                          -----
                                           \
                                            ) 5
             Zeta(-5) - Zeta(0, -5, 12) = / k
                                          -----
                                          k = 1
                             5 5
                   3.81876 10 = 3.81876 10
> subs({a=-5,m=11},E1);evalf(%);
                                           11
                                         -----
                                          \
                                           ) 1
              Zeta(5) - Zeta(0, 5, 12) = / --
                                         ----- 5
                                         k = 1 k
                   1.036913551 = 1.036913551

> subs({a=-1/2,m=11},E1);evalf(%);
                                         11
                                       -----
                                        \
                /1\ / 1 \ ) 1
            Zeta|-| - Zeta|0, -, 12| = / ------
                \2/ \ 2 / ----- (1/2)
                                       k = 1 k
                   5.322509244 = 5.322509244

> subs({a=1/2,m=11},E1);evalf(%);
                                          11
                                        -----
                                         \
               /-1\ / -1 \ ) (1/2)
           Zeta|--| - Zeta|0, --, 12| = / k
               \2 / \ 2 / -----
                                        k = 1
                   25.78490297 = 25.78490297 

-- 
G. A. Edgar                               http://www.math.ohio-state.edu/~edgar/