Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

Consider the sum

sum1 = Sum[ k/( k^7 - 2 k + 3), {k, Infinity}]
-RootSum[ 2 + 5 #1 + 21 #1^2 + 35 #1^3 + 35 #1^4 + 21 #1^5 + 7 #1^6 + #1^7 &,
          ( PolyGamma[0, -#1] + PolyGamma[0, -#1] #1)/( 5 + 42 #1 + 105 #1^2 + 140 #1^3 
             + 105 #1^4 + 42 #1^5 + 7 #1^6)& ]

Back in the day of Mathematica 3.0 this same sum gave

sum2 = RootSum[( 3 - 2 #1 + #1^7) & , -(( PolyGamma[0, -#1] #1)/(-2 + 7 #1^6))& ]
-RootSum[ 3 - 2 #1 + #1^7 &, ( PolyGamma[0, -#1] #1)/(-2 + 7 #1^6)& ]

They are numerically equal

(sum1 - sum2) // N // Chop
0
  • Are the two RootSum expressions equivalent? and If so:
  • How do I manipulate/simplify sum1 result to the simpler form of sum2?
  • Why is the current result more complicated?
share|improve this question
    
Answering your first question: those ARE numbers, not functions. So, if their numerical value is equal ... –  belisarius Aug 15 '12 at 23:28
    
Try using Normal[] on your RootSum[] objects. –  J. M. Aug 15 '12 at 23:39

1 Answer 1

PolyGamma was modified in ver.6 (see Summary of New Features in 6.0) as well as Sum last modified in ver.7 therfore the results of these sums are symbolically different. Perhaps sum1 is simpler than sum2 with respect to internal algebraic manipulations.

To verify equivalence you can make use of symbolic functionality, e.g.

RootApproximant[ sum1 - sum2 ]
0     

This may be enforced by numerical approximations to arbitrary order e.g :

N[sum1, 3000] === N[sum2, 3000]
True
share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.