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psexpand

Run Example
(%i1)cauchysum:true;
(%o1)                                true
(%i2) psexpand:multi;
(%o2)                                multi
(%i3) sumexpand:false;
(%o3)                                false
(%i4) gensumnum:0;
(%o4)                                  0
(%i5) intosum(sumcontract(sum(sum(1/((2*k)^s), k, 1, inf) + sum(1/((2*k+1)^s), k, 0, inf), s, 2,2))), simpsum;
                          inf
                          ====                      2
                          \           1          %pi
(%o5)                      >    -------------- + ----
                          /        2              24
                          ====  4 k  + 4 k + 1
                          k = 0
(%i6) 
Run Example
cauchysum:true;
(%o1)                                true
(%i2) psexpand:false;
(%o2)                                false
(%i3) simpproduct:true;
(%o3)                                true
(%i4) sumexpand:false;
(%o4)                                false
(%i5) gensumnum:1;
(%o5)                                  1
(%i6) sumcontract(intosum(sum(sum(1/((2*k)^s), k, 1, inf) + sum(1/((2*k+1)^s), k, 0, inf), s, 2,2))), simpsum;
                          inf
                          ====                      2
                          \           1          %pi
(%o6)                      >    -------------- + ----
                          /        2              24
                          ====  4 k  + 4 k + 1
                          k = 0
(%i7) 
Run Example
cauchysum:true;
(%o1)                                true
(%i2) psexpand:false;
(%o2)                                false
(%i3) simpproduct:trie;
(%o3)                                trie
(%i4) sumexpand:false;
(%o4)                                false
(%i5) gensumnum:0;
(%o5)                                  0
(%i6) sumcontract(intosum(sum(sum(1/((2*k)^s), k, 1, inf) + sum(1/((2*k+1)^s), k, 0, inf), s, 2,2)));
                         inf
                         ====
                         \          1         1
(%o6)                     >    (---------- + ----) + 1
                         /               2      2
                         ====   (2 k + 1)    4 k
                         k = 1
(%i7) 

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Help for Psexpand