731 - Numerical Summation of a Series

[lowai]

I'd like to know how to solve Numerical Summation of a Series.
Here is my thought:

Code: Select all

         infinity    1
Phi(x) =   Sigma  -------
           k = 1   k(k+x).
                                 infinity      1 - x
Let Phi1(x) =  Phi(x) - Phi(1) =  Sigma    -------------.
                                  k = 1     k(k+x)(k+1)

Let Phi2(x) = Phi1(x) / (1 - x)
  and
    Phi3(x) = Phi2(x) - Phi2(2).

Phi3(x) converges faster than Phi(x). But when x comes large, this causes
precision error.

[Alexander Denisjuk]

I haven't solved this problem yet, but you can try the following:

Code: Select all

         infinity    1
Phi(x) =   Sigma  -------
           k = 1   k(k+x).

Let Phi1n(n+alpha) =  Phi(n+alpha) - Phi(n) ,

where n is integer. |alpha|<=0.5.

I belive it will work.

[Alexander Denisjuk]

Yes, it works.

Some more hints: Phi_m(n) can be expressed by Phi_{m}(n-1) and sum_{k=1}^{N}1/k^m should be computed as sum_{k=N}^{1}1/k^m to reduce rounding errors.

[dennis]

I'm a new hand
anyone who can tell me what is brute force
and how to use it to solve 731
thx in advance